Package 'extraDistr'

Title: Additional Univariate and Multivariate Distributions
Description: Density, distribution function, quantile function and random generation for a number of univariate and multivariate distributions. This package implements the following distributions: Bernoulli, beta-binomial, beta-negative binomial, beta prime, Bhattacharjee, Birnbaum-Saunders, bivariate normal, bivariate Poisson, categorical, Dirichlet, Dirichlet-multinomial, discrete gamma, discrete Laplace, discrete normal, discrete uniform, discrete Weibull, Frechet, gamma-Poisson, generalized extreme value, Gompertz, generalized Pareto, Gumbel, half-Cauchy, half-normal, half-t, Huber density, inverse chi-squared, inverse-gamma, Kumaraswamy, Laplace, location-scale t, logarithmic, Lomax, multivariate hypergeometric, multinomial, negative hypergeometric, non-standard beta, normal mixture, Poisson mixture, Pareto, power, reparametrized beta, Rayleigh, shifted Gompertz, Skellam, slash, triangular, truncated binomial, truncated normal, truncated Poisson, Tukey lambda, Wald, zero-inflated binomial, zero-inflated negative binomial, zero-inflated Poisson.
Authors: Tymoteusz Wolodzko
Maintainer: Tymoteusz Wolodzko <[email protected]>
License: GPL-2
Version: 1.10.0
Built: 2024-11-09 05:53:21 UTC
Source: https://github.com/twolodzko/extradistr

Help Index

Bernoulli distribution

Description

Probability mass function, distribution function, quantile function and random generation for the Bernoulli distribution.

Usage

dbern(x, prob = 0.5, log = FALSE)

pbern(q, prob = 0.5, lower.tail = TRUE, log.p = FALSE)

qbern(p, prob = 0.5, lower.tail = TRUE, log.p = FALSE)

rbern(n, prob = 0.5)

Arguments

x, q

vector of quantiles.
prob

probability of success; (0 < prob < 1).
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

See Also

Binomial

Examples

prop.table(table(rbern(1e5, 0.5)))

Beta-binomial distribution

Description

Probability mass function and random generation for the beta-binomial distribution.

Usage

dbbinom(x, size, alpha = 1, beta = 1, log = FALSE)

pbbinom(q, size, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)

rbbinom(n, size, alpha = 1, beta = 1)

Arguments

x, q

vector of quantiles.
size

number of trials (zero or more).
alpha, beta

non-negative parameters of the beta distribution.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If p∼Beta(α,β)p \sim \mathrm{Beta}(\alpha, \beta) and X∼Binomial(n,p)X \sim \mathrm{Binomial}(n, p), then X∼BetaBinomial(n,α,β)X \sim \mathrm{BetaBinomial}(n, \alpha, \beta).

Probability mass function

f(x)=(nx)B(x+α,n−x+β)B(α,β)f(x) = {n \choose x} \frac{\mathrm{B}(x+\alpha, n-x+\beta)}{\mathrm{B}(\alpha, \beta)}

Cumulative distribution function is calculated using recursive algorithm that employs the fact that Γ(x)=(x−1)!\Gamma(x) = (x - 1)!, and B(x,y)=Γ(x)Γ(y)Γ(x+y)\mathrm{B}(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}, and that (nk)=∏i=1kn+1−ii{n \choose k} = \prod_{i=1}^k \frac{n+1-i}{i}. This enables re-writing probability mass function as

f(x)=(∏i=1xn+1−ii)(α+x−1)! (β+n−x−1)!(α+β+n−1)!B(α,β)f(x) = \left( \prod_{i=1}^x \frac{n+1-i}{i} \right) \frac{\frac{(\alpha+x-1)!\,(\beta+n-x-1)!}{(\alpha+\beta+n-1)!}}{\mathrm{B}(\alpha,\beta)}

what makes recursive updating from xx to x+1x+1 easy using the properties of factorials

f(x+1)=(∏i=1xn+1−ii)n+1−x+1x+1(α+x−1)! (α+x) (β+n−x−1)! (β+n−x)−1(α+β+n−1)! (α+β+n)B(α,β)f(x+1) = \left( \prod_{i=1}^x \frac{n+1-i}{i} \right) \frac{n+1-x+1}{x+1} \frac{\frac{(\alpha+x-1)! \,(\alpha+x)\,(\beta+n-x-1)! \, (\beta+n-x)^{-1}}{(\alpha+\beta+n-1)!\,(\alpha+\beta+n)}}{\mathrm{B}(\alpha,\beta)}

and let's us efficiently calculate cumulative distribution function as a sum of probability mass functions

F(x)=∑k=0xf(k)F(x) = \sum_{k=0}^x f(k)

See Also

Beta, Binomial

Examples

x <- rbbinom(1e5, 1000, 5, 13)
xx <- 0:1000
hist(x, 100, freq = FALSE)
lines(xx-0.5, dbbinom(xx, 1000, 5, 13), col = "red")
hist(pbbinom(x, 1000, 5, 13))
xx <- seq(0, 1000, by = 0.1)
plot(ecdf(x))
lines(xx, pbbinom(xx, 1000, 5, 13), col = "red", lwd = 2)

Beta-negative binomial distribution

Description

Probability mass function and random generation for the beta-negative binomial distribution.

Usage

dbnbinom(x, size, alpha = 1, beta = 1, log = FALSE)

pbnbinom(q, size, alpha = 1, beta = 1, lower.tail = TRUE, log.p = FALSE)

rbnbinom(n, size, alpha = 1, beta = 1)

Arguments

x, q

vector of quantiles.
size

number of trials (zero or more). Must be strictly positive, need not be integer.
alpha, beta

non-negative parameters of the beta distribution.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If p∼Beta(α,β)p \sim \mathrm{Beta}(\alpha, \beta) and X∼NegBinomial(r,p)X \sim \mathrm{NegBinomial}(r, p), then X∼BetaNegBinomial(r,α,β)X \sim \mathrm{BetaNegBinomial}(r, \alpha, \beta).

Probability mass function

f(x)=Γ(r+x)x! Γ(r)B(α+r,β+x)B(α,β)f(x) = \frac{\Gamma(r+x)}{x! \,\Gamma(r)} \frac{\mathrm{B}(\alpha+r, \beta+x)}{\mathrm{B}(\alpha, \beta)}

Cumulative distribution function is calculated using recursive algorithm that employs the fact that Γ(x)=(x−1)!\Gamma(x) = (x - 1)! and B(x,y)=Γ(x)Γ(y)Γ(x+y)\mathrm{B}(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. This enables re-writing probability mass function as

f(x)=(r+x−1)!x! Γ(r)(α+r−1)! (β+x−1)!(α+β+r+x−1)!B(α,β)f(x) = \frac{(r+x-1)!}{x! \, \Gamma(r)} \frac{\frac{(\alpha+r-1)!\,(\beta+x-1)!}{(\alpha+\beta+r+x-1)!}}{\mathrm{B}(\alpha,\beta)}

what makes recursive updating from xx to x+1x+1 easy using the properties of factorials

f(x+1)=(r+x−1)! (r+x)x! (x+1) Γ(r)(α+r−1)! (β+x−1)! (β+x)(α+β+r+x−1)! (α+β+r+x)B(α,β)f(x+1) = \frac{(r+x-1)!\,(r+x)}{x!\,(x+1) \, \Gamma(r)} \frac{\frac{(\alpha+r-1)!\,(\beta+x-1)!\,(\beta+x)}{(\alpha+\beta+r+x-1)!\,(\alpha+\beta+r+x)}}{\mathrm{B}(\alpha,\beta)}

and let's us efficiently calculate cumulative distribution function as a sum of probability mass functions

F(x)=∑k=0xf(k)F(x) = \sum_{k=0}^x f(k)

See Also

Beta, NegBinomial

Examples

x <- rbnbinom(1e5, 1000, 5, 13)
xx <- 0:1e5
hist(x, 100, freq = FALSE)
lines(xx-0.5, dbnbinom(xx, 1000, 5, 13), col = "red")
hist(pbnbinom(x, 1000, 5, 13))
xx <- seq(0, 1e5, by = 0.1)
plot(ecdf(x))
lines(xx, pbnbinom(xx, 1000, 5, 13), col = "red", lwd = 2)

Beta prime distribution

Description

Density, distribution function, quantile function and random generation for the beta prime distribution.

Usage

dbetapr(x, shape1, shape2, scale = 1, log = FALSE)

pbetapr(q, shape1, shape2, scale = 1, lower.tail = TRUE, log.p = FALSE)

qbetapr(p, shape1, shape2, scale = 1, lower.tail = TRUE, log.p = FALSE)

rbetapr(n, shape1, shape2, scale = 1)

Arguments

x, q

vector of quantiles.
shape1, shape2

non-negative parameters.
scale

positive valued scale parameter.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If X∼Beta(α,β)X \sim \mathrm{Beta}(\alpha, \beta), then X1−X∼BetaPrime(α,β)\frac{X}{1-X} \sim \mathrm{BetaPrime}(\alpha, \beta).

Probability density function

f(x)=(x/σ)α−1(1+x/σ)−α−βB(α,β)σf(x) = \frac{(x/\sigma)^{\alpha-1} (1+x/\sigma)^{-\alpha -\beta}}{\mathrm{B}(\alpha,\beta)\sigma}

Cumulative distribution function

F(x)=Ix/σ1+x/σ(α,β)F(x) = I_{\frac{x/\sigma}{1+x/\sigma}}(\alpha, \beta)

See Also

Beta

Examples

x <- rbetapr(1e5, 5, 3, 2)
hist(x, 350, freq = FALSE, xlim = c(0, 100))
curve(dbetapr(x, 5, 3, 2), 0, 100, col = "red", add = TRUE, n = 500)
hist(pbetapr(x, 5, 3, 2))
plot(ecdf(x), xlim = c(0, 100))
curve(pbetapr(x, 5, 3, 2), 0, 100, col = "red", add = TRUE, n = 500)

Bhattacharjee distribution

Description

Density, distribution function, and random generation for the Bhattacharjee distribution.

Usage

dbhatt(x, mu = 0, sigma = 1, a = sigma, log = FALSE)

pbhatt(q, mu = 0, sigma = 1, a = sigma, lower.tail = TRUE, log.p = FALSE)

rbhatt(n, mu = 0, sigma = 1, a = sigma)

Arguments

x, q

vector of quantiles.
mu, sigma, a

location, scale and shape parameters. Scale and shape must be positive.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If Z∼Normal(0,1)Z \sim \mathrm{Normal}(0, 1) and U∼Uniform(0,1)U \sim \mathrm{Uniform}(0, 1), then Z+UZ+U follows Bhattacharjee distribution.

Probability density function

f(z)=12a[Φ(x−μ+aσ)−Φ(x−μ−aσ)]f(z) = \frac{1}{2a} \left[\Phi\left(\frac{x-\mu+a}{\sigma}\right) - \Phi\left(\frac{x-\mu-a}{\sigma}\right)\right]

Cumulative distribution function

F(z)=σ2a[(x−μ)Φ(x−μ+aσ)−(x−μ)Φ(x−μ−aσ)+ϕ(x−μ+aσ)−ϕ(x−μ−aσ)]F(z) = \frac{\sigma}{2a} \left[(x-\mu)\Phi\left(\frac{x-\mu+a}{\sigma}\right) - (x-\mu)\Phi\left(\frac{x-\mu-a}{\sigma}\right) + \phi\left(\frac{x-\mu+a}{\sigma}\right) - \phi\left(\frac{x-\mu-a}{\sigma}\right)\right]

References

Bhattacharjee, G.P., Pandit, S.N.N., and Mohan, R. (1963). Dimensional chains involving rectangular and normal error-distributions. Technometrics, 5, 404-406.

Examples

x <- rbhatt(1e5, 5, 3, 5)
hist(x, 100, freq = FALSE)
curve(dbhatt(x, 5, 3, 5), -20, 20, col = "red", add = TRUE)
hist(pbhatt(x, 5, 3, 5))
plot(ecdf(x))
curve(pbhatt(x, 5, 3, 5), -20, 20, col = "red", lwd = 2, add = TRUE)

Birnbaum-Saunders (fatigue life) distribution

Description

Density, distribution function, quantile function and random generation for the Birnbaum-Saunders (fatigue life) distribution.

Usage

dfatigue(x, alpha, beta = 1, mu = 0, log = FALSE)

pfatigue(q, alpha, beta = 1, mu = 0, lower.tail = TRUE, log.p = FALSE)

qfatigue(p, alpha, beta = 1, mu = 0, lower.tail = TRUE, log.p = FALSE)

rfatigue(n, alpha, beta = 1, mu = 0)

Arguments

x, q

vector of quantiles.
alpha, beta, mu

shape, scale and location parameters. Scale and shape must be positive.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=(x−μβ+βx−μ2α(x−μ))ϕ(1α(x−μβ−βx−μ))f(x) = \left (\frac{\sqrt{\frac{x-\mu} {\beta}} + \sqrt{\frac{\beta} {x-\mu}}} {2\alpha (x-\mu)} \right) \phi \left( \frac{1}{\alpha}\left( \sqrt{\frac{x-\mu}{\beta}} - \sqrt{\frac{\beta}{x-\mu}} \right) \right)

Cumulative distribution function

F(x)=Φ(1α(x−μβ−βx−μ))F(x) = \Phi \left(\frac{1}{\alpha}\left( \sqrt{\frac{x-\mu}{\beta}} - \sqrt{\frac{\beta}{x-\mu}} \right) \right)

Quantile function

F−1(p)=[α2Φ−1(p)+(α2Φ−1(p))2+1]2β+μF^{-1}(p) = \left[\frac{\alpha}{2} \Phi^{-1}(p) + \sqrt{\left(\frac{\alpha}{2} \Phi^{-1}(p)\right)^{2} + 1}\right]^{2} \beta + \mu

References

Birnbaum, Z. W. and Saunders, S. C. (1969). A new family of life distributions. Journal of Applied Probability, 6(2), 637-652.

Desmond, A. (1985) Stochastic models of failure in random environments. Canadian Journal of Statistics, 13, 171-183.

Vilca-Labra, F., and Leiva-Sanchez, V. (2006). A new fatigue life model based on the family of skew-elliptical distributions. Communications in Statistics-Theory and Methods, 35(2), 229-244.

Leiva, V., Sanhueza, A., Sen, P. K., and Paula, G. A. (2008). Random number generators for the generalized Birnbaum-Saunders distribution. Journal of Statistical Computation and Simulation, 78(11), 1105-1118.

Examples

x <- rfatigue(1e5, .5, 2, 5)
hist(x, 100, freq = FALSE)
curve(dfatigue(x, .5, 2, 5), 2, 20, col = "red", add = TRUE)
hist(pfatigue(x, .5, 2, 5))
plot(ecdf(x))
curve(pfatigue(x, .5, 2, 5), 2, 20, col = "red", lwd = 2, add = TRUE)

Bivariate normal distribution

Description

Density, distribution function and random generation for the bivariate normal distribution.

Usage

dbvnorm(
  x,
  y = NULL,
  mean1 = 0,
  mean2 = mean1,
  sd1 = 1,
  sd2 = sd1,
  cor = 0,
  log = FALSE
)

rbvnorm(n, mean1 = 0, mean2 = mean1, sd1 = 1, sd2 = sd1, cor = 0)

Arguments

x, y

vectors of quantiles; alternatively x may be a two-column matrix (or data.frame) and y may be omitted.
mean1, mean2

vectors of means.
sd1, sd2

vectors of standard deviations.
cor

vector of correlations (-1 < cor < 1).
log

logical; if TRUE, probabilities p are given as log(p).
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=12π1−ρ2σ1σ2exp⁡{−12(1−ρ2)[(x1−μ1σ1)2−2ρ(x1−μ1σ1)(x2−μ2σ2)+(x2−μ2σ2)2]}f(x) = \frac{1}{2\pi\sqrt{1-\rho^2}\sigma_1\sigma_2} \exp\left\{-\frac{1}{2(1-\rho^2)} \left[\left(\frac{x_1 - \mu_1}{\sigma_1}\right)^2 - 2\rho \left(\frac{x_1 - \mu_1}{\sigma_1}\right) \left(\frac{x_2 - \mu_2}{\sigma_2}\right) + \left(\frac{x_2 - \mu_2}{\sigma_2}\right)^2\right]\right\}

References

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC

Mukhopadhyay, N. (2000). Probability and statistical inference. Chapman & Hall/CRC

See Also

Normal

Examples

y <- x <- seq(-4, 4, by = 0.25)
z <- outer(x, y, function(x, y) dbvnorm(x, y, cor = -0.75))
persp(x, y, z)

y <- x <- seq(-4, 4, by = 0.25)
z <- outer(x, y, function(x, y) dbvnorm(x, y, cor = -0.25))
persp(x, y, z)

Bivariate Poisson distribution

Description

Probability mass function and random generation for the bivariate Poisson distribution.

Usage

dbvpois(x, y = NULL, a, b, c, log = FALSE)

rbvpois(n, a, b, c)

Arguments

x, y

vectors of quantiles; alternatively x may be a two-column matrix (or data.frame) and y may be omitted.
a, b, c

positive valued parameters.
log

logical; if TRUE, probabilities p are given as log(p).
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

f(x,y)=exp⁡{−(a+b+c)}axx!byy!∑k=0min⁡(x,y)(xk)(yk)k!(cab)kf(x,y) = \exp \{-(a+b+c)\} \frac{a^x}{x!} \frac{b^y}{y!} \sum_{k=0}^{\min(x,y)} {x \choose k} {y \choose k} k! \left( \frac{c}{ab} \right)^k

References

Karlis, D. and Ntzoufras, I. (2003). Analysis of sports data by using bivariate Poisson models. Journal of the Royal Statistical Society: Series D (The Statistician), 52(3), 381-393.

Kocherlakota, S. and Kocherlakota, K. (1992) Bivariate Discrete Distributions. New York: Dekker.

Johnson, N., Kotz, S. and Balakrishnan, N. (1997). Discrete Multivariate Distributions. New York: Wiley.

Holgate, P. (1964). Estimation for the bivariate Poisson distribution. Biometrika, 51(1-2), 241-287.

Kawamura, K. (1984). Direct calculation of maximum likelihood estimator for the bivariate Poisson distribution. Kodai mathematical journal, 7(2), 211-221.

See Also

Poisson

Examples

x <- rbvpois(5000, 7, 8, 5)
image(prop.table(table(x[,1], x[,2])))
colMeans(x)

Categorical distribution

Description

Probability mass function, distribution function, quantile function and random generation for the categorical distribution.

Usage

dcat(x, prob, log = FALSE)

pcat(q, prob, lower.tail = TRUE, log.p = FALSE)

qcat(p, prob, lower.tail = TRUE, log.p = FALSE, labels)

rcat(n, prob, labels)

rcatlp(n, log_prob, labels)

Arguments

x, q

vector of quantiles.
prob, log_prob

vector of length mm, or mm-column matrix of non-negative weights (or their logarithms in log_prob).
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
labels

if provided, labeled factor vector is returned. Number of labels needs to be the same as number of categories (number of columns in prob).
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

Pr⁡(X=k)=wk∑j=1mwj\Pr(X = k) = \frac{w_k}{\sum_{j=1}^m w_j}

Cumulative distribution function

Pr⁡(X≤k)=∑i=1kwi∑j=1mwj\Pr(X \le k) = \frac{\sum_{i=1}^k w_i}{\sum_{j=1}^m w_j}

It is possible to sample from categorical distribution parametrized by vector of unnormalized log-probabilities α1,…,αm\alpha_1,\dots,\alpha_m without leaving the log space by employing the Gumbel-max trick (Maddison, Tarlow and Minka, 2014). If g1,…,gmg_1,\dots,g_m are samples from Gumbel distribution with cumulative distribution function F(g)=exp⁡(−exp⁡(−g))F(g) = \exp(-\exp(-g)), then k=arg maxi{gi+αi}k = \mathrm{arg\,max}_i \{g_i + \alpha_i\} is a draw from categorical distribution parametrized by vector of probabilities p1,…,pmp_1,\dots,p_m, such that pi=exp⁡(αi)/[∑j=1mexp⁡(αj)]p_i = \exp(\alpha_i) / [\sum_{j=1}^m \exp(\alpha_j)]. This is implemented in rcatlp function parametrized by vector of log-probabilities log_prob.

References

Maddison, C. J., Tarlow, D., & Minka, T. (2014). A* sampling. [In:] Advances in Neural Information Processing Systems (pp. 3086-3094). https://arxiv.org/abs/1411.0030

Examples

# Generating 10 random draws from categorical distribution
# with k=3 categories occuring with equal probabilities
# parametrized using a vector

rcat(10, c(1/3, 1/3, 1/3))

# or with k=5 categories parametrized using a matrix of probabilities
# (generated from Dirichlet distribution)

p <- rdirichlet(10, c(1, 1, 1, 1, 1))
rcat(10, p)

x <- rcat(1e5, c(0.2, 0.4, 0.3, 0.1))
plot(prop.table(table(x)), type = "h")
lines(0:5, dcat(0:5, c(0.2, 0.4, 0.3, 0.1)), col = "red")

p <- rdirichlet(1, rep(1, 20))
x <- rcat(1e5, matrix(rep(p, 2), nrow = 2, byrow = TRUE))
xx <- 0:21
plot(prop.table(table(x)))
lines(xx, dcat(xx, p), col = "red")

xx <- seq(0, 21, by = 0.01)
plot(ecdf(x))
lines(xx, pcat(xx, p), col = "red", lwd = 2)

pp <- seq(0, 1, by = 0.001)
plot(ecdf(x))
lines(qcat(pp, p), pp, col = "red", lwd = 2)

Dirichlet distribution

Description

Density function, cumulative distribution function and random generation for the Dirichlet distribution.

Usage

ddirichlet(x, alpha, log = FALSE)

rdirichlet(n, alpha)

Arguments

x

kk-column matrix of quantiles.
alpha

kk-values vector or kk-column matrix; concentration parameter. Must be positive.
log

logical; if TRUE, probabilities p are given as log(p).
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=Γ(∑kαk)∏kΓ(αk)∏kxkk−1f(x) = \frac{\Gamma(\sum_k \alpha_k)}{\prod_k \Gamma(\alpha_k)} \prod_k x_k^{k-1}

References

Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer-Verlag.

Examples

# Generating 10 random draws from Dirichlet distribution
# parametrized using a vector

rdirichlet(10, c(1, 1, 1, 1))

# or parametrized using a matrix where each row
# is a vector of parameters

alpha <- matrix(c(1, 1, 1, 1:3, 7:9), ncol = 3, byrow = TRUE)
rdirichlet(10, alpha)

Dirichlet-multinomial (multivariate Polya) distribution

Description

Density function, cumulative distribution function and random generation for the Dirichlet-multinomial (multivariate Polya) distribution.

Usage

ddirmnom(x, size, alpha, log = FALSE)

rdirmnom(n, size, alpha)

Arguments

x

kk-column matrix of quantiles.
size

numeric vector; number of trials (zero or more).
alpha

kk-values vector or kk-column matrix; concentration parameter. Must be positive.
log

logical; if TRUE, probabilities p are given as log(p).
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If (p1,…,pk)∼Dirichlet(α1,…,αk)(p_1,\dots,p_k) \sim \mathrm{Dirichlet}(\alpha_1,\dots,\alpha_k) and (x1,…,xk)∼Multinomial(n,p1,…,pk)(x_1,\dots,x_k) \sim \mathrm{Multinomial}(n, p_1,\dots,p_k), then (x1,…,xk)∼DirichletMultinomial(n,α1,…,αk)(x_1,\dots,x_k) \sim \mathrm{DirichletMultinomial(n, \alpha_1,\dots,\alpha_k)}.

Probability density function

f(x)=(n!)Γ(∑αk)Γ(n+∑αk)∏k=1KΓ(xk+αk)(xk!)Γ(αk)f(x) = \frac{\left(n!\right)\Gamma\left(\sum \alpha_k\right)}{\Gamma\left(n+\sum \alpha_k\right)}\prod_{k=1}^K\frac{\Gamma(x_{k}+\alpha_{k})}{\left(x_{k}!\right)\Gamma(\alpha_{k})}

References

Gentle, J.E. (2006). Random number generation and Monte Carlo methods. Springer.

Kvam, P. and Day, D. (2001) The multivariate Polya distribution in combat modeling. Naval Research Logistics, 48, 1-17.

See Also

Dirichlet, Multinomial

Discrete gamma distribution

Description

Probability mass function, distribution function and random generation for discrete gamma distribution.

Usage

ddgamma(x, shape, rate = 1, scale = 1/rate, log = FALSE)

pdgamma(q, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)

rdgamma(n, shape, rate = 1, scale = 1/rate)

Arguments

x, q

vector of quantiles.
shape, scale

shape and scale parameters. Must be positive, scale strictly.
rate

an alternative way to specify the scale.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function of discrete gamma distribution fY(y)f_Y(y) is defined by discretization of continuous gamma distribution fY(y)=SX(y)−SX(y+1)f_Y(y) = S_X(y) - S_X(y+1) where SXS_X is a survival function of continuous gamma distribution.

References

Chakraborty, S. and Chakravarty, D. (2012). Discrete Gamma distributions: Properties and parameter estimations. Communications in Statistics-Theory and Methods, 41(18), 3301-3324.

See Also

GammaDist, DiscreteNormal

Examples

x <- rdgamma(1e5, 9, 1)
xx <- 0:50
plot(prop.table(table(x)))
lines(xx, ddgamma(xx, 9, 1), col = "red")
hist(pdgamma(x, 9, 1))
plot(ecdf(x))
xx <- seq(0, 50, 0.1)
lines(xx, pdgamma(xx, 9, 1), col = "red", lwd = 2, type = "s")

Discrete Laplace distribution

Description

Probability mass, distribution function and random generation for the discrete Laplace distribution parametrized by location and scale.

Usage

ddlaplace(x, location, scale, log = FALSE)

pdlaplace(q, location, scale, lower.tail = TRUE, log.p = FALSE)

rdlaplace(n, location, scale)

Arguments

x, q

vector of quantiles.
location

location parameter.
scale

scale parameter; 0 < scale < 1.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If U∼Geometric(1−p)U \sim \mathrm{Geometric}(1-p) and V∼Geometric(1−p)V \sim \mathrm{Geometric}(1-p), then U−V∼DiscreteLaplace(p)U-V \sim \mathrm{DiscreteLaplace}(p), where geometric distribution is related to discrete Laplace distribution in similar way as exponential distribution is related to Laplace distribution.

Probability mass function

f(x)=1−p1+pp∣x−μ∣f(x) = \frac{1-p}{1+p} p^{|x-\mu|}

Cumulative distribution function

F(x)={p−∣x−μ∣1+px<01−p∣x−μ∣+11+px≥0F(x) = \left\{\begin{array}{ll} \frac{p^{-|x-\mu|}}{1+p} & x < 0 \\ 1 - \frac{p^{|x-\mu|+1}}{1+p} & x \ge 0 \end{array}\right.

References

Inusah, S., & Kozubowski, T.J. (2006). A discrete analogue of the Laplace distribution. Journal of statistical planning and inference, 136(3), 1090-1102.

Kotz, S., Kozubowski, T., & Podgorski, K. (2012). The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Springer Science & Business Media.

Examples

p <- 0.45
x <- rdlaplace(1e5, 0, p)
xx <- seq(-200, 200, by = 1)
plot(prop.table(table(x)))
lines(xx, ddlaplace(xx, 0, p), col = "red")
hist(pdlaplace(x, 0, p))
plot(ecdf(x))
lines(xx, pdlaplace(xx, 0, p), col = "red", type = "s")

Discrete normal distribution

Description

Probability mass function, distribution function and random generation for discrete normal distribution.

Usage

ddnorm(x, mean = 0, sd = 1, log = FALSE)

pdnorm(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

rdnorm(n, mean = 0, sd = 1)

Arguments

x, q

vector of quantiles.
mean

vector of means.
sd

vector of standard deviations.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

f(x)=Φ(x−μ+1σ)−Φ(x−μσ)f(x) = \Phi\left(\frac{x-\mu+1}{\sigma}\right) - \Phi\left(\frac{x-\mu}{\sigma}\right)

Cumulative distribution function

F(x)=Φ(⌊x⌋+1−μσ)F(x) = \Phi\left(\frac{\lfloor x \rfloor + 1 - \mu}{\sigma}\right)

References

Roy, D. (2003). The discrete normal distribution. Communications in Statistics-Theory and Methods, 32, 1871-1883.

See Also

Normal

Examples

x <- rdnorm(1e5, 0, 3)
xx <- -15:15
plot(prop.table(table(x)))
lines(xx, ddnorm(xx, 0, 3), col = "red")
hist(pdnorm(x, 0, 3))
plot(ecdf(x))
xx <- seq(-15, 15, 0.1)
lines(xx, pdnorm(xx, 0, 3), col = "red", lwd = 2, type = "s")

Discrete uniform distribution

Description

Probability mass function, distribution function, quantile function and random generation for the discrete uniform distribution.

Usage

ddunif(x, min, max, log = FALSE)

pdunif(q, min, max, lower.tail = TRUE, log.p = FALSE)

qdunif(p, min, max, lower.tail = TRUE, log.p = FALSE)

rdunif(n, min, max)

Arguments

x, q

vector of quantiles.
min, max

lower and upper limits of the distribution. Must be finite.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If min == max, then discrete uniform distribution is a degenerate distribution.

Examples

x <- rdunif(1e5, 1, 10) 
xx <- -1:11
plot(prop.table(table(x)), type = "h")
lines(xx, ddunif(xx, 1, 10), col = "red")
hist(pdunif(x, 1, 10))
xx <- seq(-1, 11, by = 0.01)
plot(ecdf(x))
lines(xx, pdunif(xx, 1, 10), col = "red")

Discrete Weibull distribution (type I)

Description

Density, distribution function, quantile function and random generation for the discrete Weibull (type I) distribution.

Usage

ddweibull(x, shape1, shape2, log = FALSE)

pdweibull(q, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

qdweibull(p, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

rdweibull(n, shape1, shape2)

Arguments

x, q

vector of quantiles.
shape1, shape2

parameters (named q, β\beta). Values of shape2 need to be positive and 0 < shape1 < 1.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

f(x)=qxβ−q(x+1)βf(x) = q^{x^\beta} - q^{(x+1)^\beta}

Cumulative distribution function

F(x)=1−q(x+1)βF(x) = 1-q^{(x+1)^\beta}

Quantile function

F−1(p)=⌈(log⁡(1−p)log⁡(q))1/β−1⌉F^{-1}(p) = \left \lceil{\left(\frac{\log(1-p)}{\log(q)}\right)^{1/\beta} - 1}\right \rceil

References

Nakagawa, T. and Osaki, S. (1975). The Discrete Weibull Distribution. IEEE Transactions on Reliability, R-24, 300-301.

Kulasekera, K.B. (1994). Approximate MLE's of the parameters of a discrete Weibull distribution with type I censored data. Microelectronics Reliability, 34(7), 1185-1188.

Khan, M.A., Khalique, A. and Abouammoh, A.M. (1989). On estimating parameters in a discrete Weibull distribution. IEEE Transactions on Reliability, 38(3), 348-350.

See Also

Weibull

Examples

x <- rdweibull(1e5, 0.32, 1)
xx <- seq(-2, 100, by = 1)
plot(prop.table(table(x)), type = "h")
lines(xx, ddweibull(xx, .32, 1), col = "red")

# Notice: distribution of F(X) is far from uniform:
hist(pdweibull(x, .32, 1), 50)

plot(ecdf(x))
lines(xx, pdweibull(xx, .32, 1), col = "red", lwd = 2, type = "s")

Frechet distribution

Description

Density, distribution function, quantile function and random generation for the Frechet distribution.

Usage

dfrechet(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

pfrechet(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qfrechet(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rfrechet(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q

vector of quantiles.
lambda, sigma, mu

shape, scale, and location parameters. Scale and shape must be positive.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=λσ(x−μσ)−1−λexp⁡(−(x−μσ)−λ)f(x) = \frac{\lambda}{\sigma} \left(\frac{x-\mu}{\sigma}\right)^{-1-\lambda} \exp\left(-\left(\frac{x-\mu}{\sigma}\right)^{-\lambda}\right)

Cumulative distribution function

F(x)=exp⁡(−(x−μσ)−λ)F(x) = \exp\left(-\left(\frac{x-\mu}{\sigma}\right)^{-\lambda}\right)

Quantile function

F−1(p)=μ+σ−log⁡(p)−1/λF^{-1}(p) = \mu + \sigma -\log(p)^{-1/\lambda}

References

Bury, K. (1999). Statistical Distributions in Engineering. Cambridge University Press.

Examples

x <- rfrechet(1e5, 5, 2, 1.5)
xx <- seq(0, 1000, by = 0.1)
hist(x, 200, freq = FALSE)
lines(xx, dfrechet(xx, 5, 2, 1.5), col = "red") 
hist(pfrechet(x, 5, 2, 1.5))
plot(ecdf(x))
lines(xx, pfrechet(xx, 5, 2, 1.5), col = "red", lwd = 2)

Gamma-Poisson distribution

Description

Probability mass function and random generation for the gamma-Poisson distribution.

Usage

dgpois(x, shape, rate, scale = 1/rate, log = FALSE)

pgpois(q, shape, rate, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)

rgpois(n, shape, rate, scale = 1/rate)

Arguments

x, q

vector of quantiles.
shape, scale

shape and scale parameters. Must be positive, scale strictly.
rate

an alternative way to specify the scale.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Gamma-Poisson distribution arises as a continuous mixture of Poisson distributions, where the mixing distribution of the Poisson rate λ\lambda is a gamma distribution. When X∼Poisson(λ)X \sim \mathrm{Poisson}(\lambda) and λ∼Gamma(α,β)\lambda \sim \mathrm{Gamma}(\alpha, \beta), then X∼GammaPoisson(α,β)X \sim \mathrm{GammaPoisson}(\alpha, \beta).

Probability mass function

f(x)=Γ(α+x)x! Γ(α)(β1+β)x(1−β1+β)αf(x) = \frac{\Gamma(\alpha+x)}{x! \, \Gamma(\alpha)} \left(\frac{\beta}{1+\beta}\right)^x \left(1-\frac{\beta}{1+\beta}\right)^\alpha

Cumulative distribution function is calculated using recursive algorithm that employs the fact that Γ(x)=(x−1)!\Gamma(x) = (x - 1)!. This enables re-writing probability mass function as

f(x)=(α+x−1)!x! Γ(α)(β1+β)x(1−β1+β)αf(x) = \frac{(\alpha+x-1)!}{x! \, \Gamma(\alpha)} \left( \frac{\beta}{1+\beta} \right)^x \left( 1- \frac{\beta}{1+\beta} \right)^\alpha

what makes recursive updating from xx to x+1x+1 easy using the properties of factorials

f(x+1)=(α+x−1)! (α+x)x! (x+1) Γ(α)(β1+β)x(β1+β)(1−β1+β)αf(x+1) = \frac{(\alpha+x-1)! \, (\alpha+x)}{x! \,(x+1) \, \Gamma(\alpha)} \left( \frac{\beta}{1+\beta} \right)^x \left( \frac{\beta}{1+\beta} \right) \left( 1- \frac{\beta}{1+\beta} \right)^\alpha

and let's us efficiently calculate cumulative distribution function as a sum of probability mass functions

F(x)=∑k=0xf(k)F(x) = \sum_{k=0}^x f(k)

See Also

Gamma, Poisson

Examples

x <- rgpois(1e5, 7, 0.002)
xx <- seq(0, 12000, by = 1)
hist(x, 100, freq = FALSE)
lines(xx, dgpois(xx, 7, 0.002), col = "red")
hist(pgpois(x, 7, 0.002))
xx <- seq(0, 12000, by = 0.1)
plot(ecdf(x))
lines(xx, pgpois(xx, 7, 0.002), col = "red", lwd = 2)

Generalized extreme value distribution

Description

Density, distribution function, quantile function and random generation for the generalized extreme value distribution.

Usage

dgev(x, mu = 0, sigma = 1, xi = 0, log = FALSE)

pgev(q, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

qgev(p, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

rgev(n, mu = 0, sigma = 1, xi = 0)

Arguments

x, q

vector of quantiles.
mu, sigma, xi

location, scale, and shape parameters. Scale must be positive.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)={1σ(1+ξx−μσ)−1/ξ−1exp⁡(−(1+ξx−μσ)−1/ξ)ξ≠01σexp⁡(−x−μσ)exp⁡(−exp⁡(−x−μσ))ξ=0f(x) = \left\{\begin{array}{ll} \frac{1}{\sigma} \left(1+\xi \frac{x-\mu}{\sigma}\right)^{-1/\xi-1} \exp\left(-\left(1+\xi \frac{x-\mu}{\sigma}\right)^{-1/\xi}\right) & \xi \neq 0 \\ \frac{1}{\sigma} \exp\left(- \frac{x-\mu}{\sigma}\right) \exp\left(-\exp\left(- \frac{x-\mu}{\sigma}\right)\right) & \xi = 0 \end{array}\right.

Cumulative distribution function

F(x)={exp⁡(−(1+ξx−μσ)1/ξ)ξ≠0exp⁡(−exp⁡(−x−μσ))ξ=0F(x) = \left\{\begin{array}{ll} \exp\left(-\left(1+\xi \frac{x-\mu}{\sigma}\right)^{1/\xi}\right) & \xi \neq 0 \\ \exp\left(-\exp\left(- \frac{x-\mu}{\sigma}\right)\right) & \xi = 0 \end{array}\right.

Quantile function

F−1(p)={μ−σξ(1−(−log⁡(p))ξ)ξ≠0μ−σlog⁡(−log⁡(p))ξ=0F^{-1}(p) = \left\{\begin{array}{ll} \mu - \frac{\sigma}{\xi} (1 - (-\log(p))^\xi) & \xi \neq 0 \\ \mu - \sigma \log(-\log(p)) & \xi = 0 \end{array}\right.

References

Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer.

Examples

curve(dgev(x, xi = -1/2), -4, 4, col = "green", ylab = "")
curve(dgev(x, xi = 0), -4, 4, col = "red", add = TRUE)
curve(dgev(x, xi = 1/2), -4, 4, col = "blue", add = TRUE)
legend("topleft", col = c("green", "red", "blue"), lty = 1,
       legend = expression(xi == -1/2, xi == 0, xi == 1/2), bty = "n")

x <- rgev(1e5, 5, 2, .5)
hist(x, 1000, freq = FALSE, xlim = c(0, 50))
curve(dgev(x, 5, 2, .5), 0, 50, col = "red", add = TRUE, n = 5000)
hist(pgev(x, 5, 2, .5))
plot(ecdf(x), xlim = c(0, 50))
curve(pgev(x, 5, 2, .5), 0, 50, col = "red", lwd = 2, add = TRUE)

Gompertz distribution

Description

Density, distribution function, quantile function and random generation for the Gompertz distribution.

Usage

dgompertz(x, a = 1, b = 1, log = FALSE)

pgompertz(q, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)

qgompertz(p, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)

rgompertz(n, a = 1, b = 1)

Arguments

x, q

vector of quantiles.
a, b

positive valued scale and location parameters.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=aexp⁡(bx−ab(exp⁡(bx)−1))f(x) = a \exp\left(bx - \frac{a}{b} (\exp(bx)-1)\right)

Cumulative distribution function

F(x)=1−exp⁡(−ab(exp⁡(bx)−1))F(x) = 1-\exp\left(-\frac{a}{b} (\exp(bx)-1)\right)

Quantile function

F−1(p)=1blog⁡(1−balog⁡(1−p))F^{-1}(p) = \frac{1}{b} \log\left(1-\frac{b}{a}\log(1-p)\right)

References

Lenart, A. (2012). The Gompertz distribution and Maximum Likelihood Estimation of its parameters - a revision. MPIDR WORKING PAPER WP 2012-008. https://www.demogr.mpg.de/papers/working/wp-2012-008.pdf

Examples

x <- rgompertz(1e5, 5, 2)
hist(x, 100, freq = FALSE)
curve(dgompertz(x, 5, 2), 0, 1, col = "red", add = TRUE)
hist(pgompertz(x, 5, 2))
plot(ecdf(x))
curve(pgompertz(x, 5, 2), 0, 1, col = "red", lwd = 2, add = TRUE)

Generalized Pareto distribution

Description

Density, distribution function, quantile function and random generation for the generalized Pareto distribution.

Usage

dgpd(x, mu = 0, sigma = 1, xi = 0, log = FALSE)

pgpd(q, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

qgpd(p, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

rgpd(n, mu = 0, sigma = 1, xi = 0)

Arguments

x, q

vector of quantiles.
mu, sigma, xi

location, scale, and shape parameters. Scale must be positive.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)={1σ(1+ξx−μσ)−(ξ+1)/ξξ≠01σexp⁡(−x−μσ)ξ=0f(x) = \left\{\begin{array}{ll} \frac{1}{\sigma} \left(1+\xi \frac{x-\mu}{\sigma}\right)^{-(\xi+1)/\xi} & \xi \neq 0 \\ \frac{1}{\sigma} \exp\left(-\frac{x-\mu}{\sigma}\right) & \xi = 0 \end{array}\right.

Cumulative distribution function

F(x)={1−(1+ξx−μσ)−1/ξξ≠01−exp⁡(−x−μσ)ξ=0F(x) = \left\{\begin{array}{ll} 1-\left(1+\xi \frac{x-\mu}{\sigma}\right)^{-1/\xi} & \xi \neq 0 \\ 1-\exp\left(-\frac{x-\mu}{\sigma}\right) & \xi = 0 \end{array}\right.

Quantile function

F−1(x)={μ+σ(1−p)−ξ−1ξξ≠0μ−σlog⁡(1−p)ξ=0F^{-1}(x) = \left\{\begin{array}{ll} \mu + \sigma \frac{(1-p)^{-\xi}-1}{\xi} & \xi \neq 0 \\ \mu - \sigma \log(1-p) & \xi = 0 \end{array}\right.

References

Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer.

Examples

x <- rgpd(1e5, 5, 2, .1)
hist(x, 100, freq = FALSE, xlim = c(0, 50))
curve(dgpd(x, 5, 2, .1), 0, 50, col = "red", add = TRUE, n = 5000)
hist(pgpd(x, 5, 2, .1))
plot(ecdf(x))
curve(pgpd(x, 5, 2, .1), 0, 50, col = "red", lwd = 2, add = TRUE)

Gumbel distribution

Description

Density, distribution function, quantile function and random generation for the Gumbel distribution.

Usage

dgumbel(x, mu = 0, sigma = 1, log = FALSE)

pgumbel(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qgumbel(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rgumbel(n, mu = 0, sigma = 1)

Arguments

x, q

vector of quantiles.
mu, sigma

location and scale parameters. Scale must be positive.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=1σexp⁡(−(x−μσ+exp⁡(−x−μσ)))f(x) = \frac{1}{\sigma} \exp\left(-\left(\frac{x-\mu}{\sigma} + \exp\left(-\frac{x-\mu}{\sigma}\right)\right)\right)

Cumulative distribution function

F(x)=exp⁡(−exp⁡(−x−μσ))F(x) = \exp\left(-\exp\left(-\frac{x-\mu}{\sigma}\right)\right)

Quantile function

F−1(p)=μ−σlog⁡(−log⁡(p))F^{-1}(p) = \mu - \sigma \log(-\log(p))

References

Bury, K. (1999). Statistical Distributions in Engineering. Cambridge University Press.

Examples

x <- rgumbel(1e5, 5, 2)
hist(x, 100, freq = FALSE)
curve(dgumbel(x, 5, 2), 0, 25, col = "red", add = TRUE)
hist(pgumbel(x, 5, 2))
plot(ecdf(x))
curve(pgumbel(x, 5, 2), 0, 25, col = "red", lwd = 2, add = TRUE)

Half-Cauchy distribution

Description

Density, distribution function, quantile function and random generation for the half-Cauchy distribution.

Usage

dhcauchy(x, sigma = 1, log = FALSE)

phcauchy(q, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qhcauchy(p, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rhcauchy(n, sigma = 1)

Arguments

x, q

vector of quantiles.
sigma

positive valued scale parameter.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If XX follows Cauchy centered at 0 and parametrized by scale σ\sigma, then ∣X∣|X| follows half-Cauchy distribution parametrized by scale σ\sigma. Half-Cauchy distribution is a special case of half-t distribution with ν=1\nu=1 degrees of freedom.

References

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian analysis, 1(3), 515-534.

Jacob, E. and Jayakumar, K. (2012). On Half-Cauchy Distribution and Process. International Journal of Statistika and Mathematika, 3(2), 77-81.

See Also

HalfT

Examples

x <- rhcauchy(1e5, 2)
hist(x, 2e5, freq = FALSE, xlim = c(0, 100))
curve(dhcauchy(x, 2), 0, 100, col = "red", add = TRUE)
hist(phcauchy(x, 2))
plot(ecdf(x), xlim = c(0, 100))
curve(phcauchy(x, 2), col = "red", lwd = 2, add = TRUE)

Half-normal distribution

Description

Density, distribution function, quantile function and random generation for the half-normal distribution.

Usage

dhnorm(x, sigma = 1, log = FALSE)

phnorm(q, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qhnorm(p, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rhnorm(n, sigma = 1)

Arguments

x, q

vector of quantiles.
sigma

positive valued scale parameter.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If XX follows normal distribution centered at 0 and parametrized by scale σ\sigma, then ∣X∣|X| follows half-normal distribution parametrized by scale σ\sigma. Half-t distribution with ν=∞\nu=\infty degrees of freedom converges to half-normal distribution.

References

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian analysis, 1(3), 515-534.

Jacob, E. and Jayakumar, K. (2012). On Half-Cauchy Distribution and Process. International Journal of Statistika and Mathematika, 3(2), 77-81.

See Also

HalfT

Examples

x <- rhnorm(1e5, 2)
hist(x, 100, freq = FALSE)
curve(dhnorm(x, 2), 0, 8, col = "red", add = TRUE)
hist(phnorm(x, 2))
plot(ecdf(x))
curve(phnorm(x, 2), 0, 8, col = "red", lwd = 2, add = TRUE)

Half-t distribution

Description

Density, distribution function, quantile function and random generation for the half-t distribution.

Usage

dht(x, nu, sigma = 1, log = FALSE)

pht(q, nu, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qht(p, nu, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rht(n, nu, sigma = 1)

Arguments

x, q

vector of quantiles.
nu, sigma

positive valued degrees of freedom and scale parameters.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If XX follows t distribution parametrized by degrees of freedom ν\nu and scale σ\sigma, then ∣X∣|X| follows half-t distribution parametrized by degrees of freedom ν\nu and scale σ\sigma.

References

Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian analysis, 1(3), 515-534.

Jacob, E. and Jayakumar, K. (2012). On Half-Cauchy Distribution and Process. International Journal of Statistika and Mathematika, 3(2), 77-81.

See Also

HalfNormal, HalfCauchy

Examples

x <- rht(1e5, 2, 2)
hist(x, 500, freq = FALSE, xlim = c(0, 100))
curve(dht(x, 2, 2), 0, 100, col = "red", add = TRUE)
hist(pht(x, 2, 2))
plot(ecdf(x), xlim = c(0, 100))
curve(pht(x, 2, 2), 0, 100, col = "red", lwd = 2, add = TRUE)

"Huber density" distribution

Description

Density, distribution function, quantile function and random generation for the "Huber density" distribution.

Usage

dhuber(x, mu = 0, sigma = 1, epsilon = 1.345, log = FALSE)

phuber(q, mu = 0, sigma = 1, epsilon = 1.345, lower.tail = TRUE, log.p = FALSE)

qhuber(p, mu = 0, sigma = 1, epsilon = 1.345, lower.tail = TRUE, log.p = FALSE)

rhuber(n, mu = 0, sigma = 1, epsilon = 1.345)

Arguments

x, q

vector of quantiles.
mu, sigma, epsilon

location, and scale, and shape parameters. Scale and shape must be positive.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Huber density is connected to Huber loss and can be defined as:

f(x)=122π(Φ(k)+ϕ(k)/k−12)e−ρk(x)f(x) = \frac{1}{2 \sqrt{2\pi} \left( \Phi(k) + \phi(k)/k - \frac{1}{2} \right)} e^{-\rho_k(x)}

where

ρk(x)={12x2∣x∣≤kk∣x∣−12k2∣x∣>k\rho_k(x) = \left\{\begin{array}{ll} \frac{1}{2} x^2 & |x|\le k \\ k|x|- \frac{1}{2} k^2 & |x|>k \end{array}\right.

References

Huber, P.J. (1964). Robust Estimation of a Location Parameter. Annals of Statistics, 53(1), 73-101.

Huber, P.J. (1981). Robust Statistics. Wiley.

Schumann, D. (2009). Robust Variable Selection. ProQuest.

Examples

x <- rhuber(1e5, 5, 2, 3)
hist(x, 100, freq = FALSE)
curve(dhuber(x, 5, 2, 3), -20, 20, col = "red", add = TRUE, n = 5000)
hist(phuber(x, 5, 2, 3))
plot(ecdf(x))
curve(phuber(x, 5, 2, 3), -20, 20, col = "red", lwd = 2, add = TRUE)

Inverse chi-squared and scaled chi-squared distributions

Description

Density, distribution function and random generation for the inverse chi-squared distribution and scaled chi-squared distribution.

Usage

dinvchisq(x, nu, tau, log = FALSE)

pinvchisq(q, nu, tau, lower.tail = TRUE, log.p = FALSE)

qinvchisq(p, nu, tau, lower.tail = TRUE, log.p = FALSE)

rinvchisq(n, nu, tau)

Arguments

x, q

vector of quantiles.
nu

positive valued shape parameter.
tau

positive valued scaling parameter; if provided it returns values for scaled chi-squared distributions.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If XX follows χ2(ν)\chi^2 (\nu) distribution, then 1/X1/X follows inverse chi-squared distribution parametrized by ν\nu. Inverse chi-squared distribution is a special case of inverse gamma distribution with parameters α=ν2\alpha=\frac{\nu}{2} and β=12\beta=\frac{1}{2}; or α=ν2\alpha=\frac{\nu}{2} and β=ντ22\beta=\frac{\nu\tau^2}{2} for scaled inverse chi-squared distribution.

See Also

Chisquare, GammaDist

Examples

x <- rinvchisq(1e5, 20)
hist(x, 100, freq = FALSE)
curve(dinvchisq(x, 20), 0, 1, n = 501, col = "red", add = TRUE)
hist(pinvchisq(x, 20))
plot(ecdf(x))
curve(pinvchisq(x, 20), 0, 1, n = 501, col = "red", lwd = 2, add = TRUE)

# scaled

x <- rinvchisq(1e5, 10, 5)
hist(x, 100, freq = FALSE)
curve(dinvchisq(x, 10, 5), 0, 150, n = 501, col = "red", add = TRUE)
hist(pinvchisq(x, 10, 5))
plot(ecdf(x))
curve(pinvchisq(x, 10, 5), 0, 150, n = 501, col = "red", lwd = 2, add = TRUE)

Inverse-gamma distribution

Description

Density, distribution function and random generation for the inverse-gamma distribution.

Usage

dinvgamma(x, alpha, beta = 1, log = FALSE)

pinvgamma(q, alpha, beta = 1, lower.tail = TRUE, log.p = FALSE)

qinvgamma(p, alpha, beta = 1, lower.tail = TRUE, log.p = FALSE)

rinvgamma(n, alpha, beta = 1)

Arguments

x, q

vector of quantiles.
alpha, beta

positive valued shape and scale parameters.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

f(x)=βαx−α−1exp⁡(−βx)Γ(α)f(x) = \frac{\beta^\alpha x^{-\alpha-1} \exp(-\frac{\beta}{x})}{\Gamma(\alpha)}

Cumulative distribution function

F(x)=γ(α,βx)Γ(α)F(x) = \frac{\gamma(\alpha, \frac{\beta}{x})}{\Gamma(\alpha)}

References

Witkovsky, V. (2001). Computing the distribution of a linear combination of inverted gamma variables. Kybernetika 37(1), 79-90.

Leemis, L.M. and McQueston, L.T. (2008). Univariate Distribution Relationships. American Statistician 62(1): 45-53.

See Also

GammaDist

Examples

x <- rinvgamma(1e5, 20, 3)
hist(x, 100, freq = FALSE)
curve(dinvgamma(x, 20, 3), 0, 1, col = "red", add = TRUE, n = 5000)
hist(pinvgamma(x, 20, 3))
plot(ecdf(x))
curve(pinvgamma(x, 20, 3), 0, 1, col = "red", lwd = 2, add = TRUE, n = 5000)

Kumaraswamy distribution

Description

Density, distribution function, quantile function and random generation for the Kumaraswamy distribution.

Usage

dkumar(x, a = 1, b = 1, log = FALSE)

pkumar(q, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)

qkumar(p, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)

rkumar(n, a = 1, b = 1)

Arguments

x, q

vector of quantiles.
a, b

positive valued parameters.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=abxa−1(1−xa)b−1f(x) = abx^{a-1} (1-x^a)^{b-1}

Cumulative distribution function

F(x)=1−(1−xa)bF(x) = 1-(1-x^a)^b

Quantile function

F−1(p)=1−(1−p1/b)1/aF^{-1}(p) = 1-(1-p^{1/b})^{1/a}

References

Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6, 70-81.

Cordeiro, G.M. and de Castro, M. (2009). A new family of generalized distributions. Journal of Statistical Computation & Simulation, 1-17.

Examples

x <- rkumar(1e5, 5, 16)
hist(x, 100, freq = FALSE)
curve(dkumar(x, 5, 16), 0, 1, col = "red", add = TRUE)
hist(pkumar(x, 5, 16))
plot(ecdf(x))
curve(pkumar(x, 5, 16), 0, 1, col = "red", lwd = 2, add = TRUE)

Laplace distribution

Description

Density, distribution function, quantile function and random generation for the Laplace distribution.

Usage

dlaplace(x, mu = 0, sigma = 1, log = FALSE)

plaplace(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qlaplace(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rlaplace(n, mu = 0, sigma = 1)

Arguments

x, q

vector of quantiles.
mu, sigma

location and scale parameters. Scale must be positive.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=12σexp⁡(−∣x−μσ∣)f(x) = \frac{1}{2\sigma} \exp\left(-\left|\frac{x-\mu}{\sigma}\right|\right)

Cumulative distribution function

F(x)={12exp⁡(x−μσ)x<μ1−12exp⁡(x−μσ)x≥μF(x) = \left\{\begin{array}{ll} \frac{1}{2} \exp\left(\frac{x-\mu}{\sigma}\right) & x < \mu \\ 1 - \frac{1}{2} \exp\left(\frac{x-\mu}{\sigma}\right) & x \geq \mu \end{array}\right.

Quantile function

F−1(p)={μ+σlog⁡(2p)p<0.5μ−σlog⁡(2(1−p))p≥0.5F^{-1}(p) = \left\{\begin{array}{ll} \mu + \sigma \log(2p) & p < 0.5 \\ \mu - \sigma \log(2(1-p)) & p \geq 0.5 \end{array}\right.

References

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC

Forbes, C., Evans, M. Hastings, N., & Peacock, B. (2011). Statistical Distributions. John Wiley & Sons.

Examples

x <- rlaplace(1e5, 5, 16)
hist(x, 100, freq = FALSE)
curve(dlaplace(x, 5, 16), -200, 200, n = 500, col = "red", add = TRUE)
hist(plaplace(x, 5, 16))
plot(ecdf(x))
curve(plaplace(x, 5, 16), -200, 200, n = 500, col = "red", lwd = 2, add = TRUE)

Location-scale version of the t-distribution

Description

Probability mass function, distribution function and random generation for location-scale version of the t-distribution. Location-scale version of the t-distribution besides degrees of freedom ν\nu, is parametrized using additional parameters μ\mu for location and σ\sigma for scale (μ=0\mu = 0 and σ=1\sigma = 1 for standard t-distribution).

Usage

dlst(x, df, mu = 0, sigma = 1, log = FALSE)

plst(q, df, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qlst(p, df, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rlst(n, df, mu = 0, sigma = 1)

Arguments

x, q

vector of quantiles.
df

degrees of freedom (> 0, maybe non-integer). df = Inf is allowed.
mu

vector of locations
sigma

vector of positive valued scale parameters.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

See Also

TDist

Examples

x <- rlst(1e5, 1000, 5, 13)
hist(x, 100, freq = FALSE)
curve(dlst(x, 1000, 5, 13), -60, 60, col = "red", add = TRUE)
hist(plst(x, 1000, 5, 13))
plot(ecdf(x))
curve(plst(x, 1000, 5, 13), -60, 60, col = "red", lwd = 2, add = TRUE)

Logarithmic series distribution

Description

Density, distribution function, quantile function and random generation for the logarithmic series distribution.

Usage

dlgser(x, theta, log = FALSE)

plgser(q, theta, lower.tail = TRUE, log.p = FALSE)

qlgser(p, theta, lower.tail = TRUE, log.p = FALSE)

rlgser(n, theta)

Arguments

x, q

vector of quantiles.
theta

vector; concentration parameter; (0 < theta < 1).
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

f(x)=−1log⁡(1−θ)θxxf(x) = \frac{-1}{\log(1-\theta)} \frac{\theta^x}{x}

Cumulative distribution function

F(x)=−1log⁡(1−θ)∑k=1xθxxF(x) = \frac{-1}{\log(1-\theta)} \sum_{k=1}^x \frac{\theta^x}{x}

Quantile function and random generation are computed using algorithm described in Krishnamoorthy (2006).

References

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC

Forbes, C., Evans, M. Hastings, N., & Peacock, B. (2011). Statistical Distributions. John Wiley & Sons.

Examples

x <- rlgser(1e5, 0.66)
xx <- seq(0, 100, by = 1)
plot(prop.table(table(x)), type = "h")
lines(xx, dlgser(xx, 0.66), col = "red")

# Notice: distribution of F(X) is far from uniform:
hist(plgser(x, 0.66), 50)

xx <- seq(0, 100, by = 0.01)
plot(ecdf(x))
lines(xx, plgser(xx, 0.66), col = "red", lwd = 2)

Lomax distribution

Description

Density, distribution function, quantile function and random generation for the Lomax distribution.

Usage

dlomax(x, lambda, kappa, log = FALSE)

plomax(q, lambda, kappa, lower.tail = TRUE, log.p = FALSE)

qlomax(p, lambda, kappa, lower.tail = TRUE, log.p = FALSE)

rlomax(n, lambda, kappa)

Arguments

x, q

vector of quantiles.
lambda, kappa

positive valued parameters.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=λκ(1+λx)κ+1f(x) = \frac{\lambda \kappa}{(1+\lambda x)^{\kappa+1}}

Cumulative distribution function

F(x)=1−(1+λx)−κF(x) = 1-(1+\lambda x)^{-\kappa}

Quantile function

F−1(p)=(1−p)−1/κ−1λF^{-1}(p) = \frac{(1-p)^{-1/\kappa} -1}{\lambda}

Examples

x <- rlomax(1e5, 5, 16)
hist(x, 100, freq = FALSE)
curve(dlomax(x, 5, 16), 0, 1, col = "red", add = TRUE, n = 5000)
hist(plomax(x, 5, 16))
plot(ecdf(x))
curve(plomax(x, 5, 16), 0, 1, col = "red", lwd = 2, add = TRUE)

Multivariate hypergeometric distribution

Description

Probability mass function and random generation for the multivariate hypergeometric distribution.

Usage

dmvhyper(x, n, k, log = FALSE)

rmvhyper(nn, n, k)

Arguments

x

mm-column matrix of quantiles.
n

mm-length vector or mm-column matrix of numbers of balls in mm colors.
k

the number of balls drawn from the urn.
log

logical; if TRUE, probabilities p are given as log(p).
nn

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

f(x)=∏i=1m(nixi)(Nk)f(x) = \frac{\prod_{i=1}^m {n_i \choose x_i}}{{N \choose k}}

The multivariate hypergeometric distribution is generalization of hypergeometric distribution. It is used for sampling without replacement kk out of NN marbles in mm colors, where each of the colors appears nin_i times. Where k=∑i=1mxik=\sum_{i=1}^m x_i, N=∑i=1mniN=\sum_{i=1}^m n_i and k≤Nk \le N.

References

Gentle, J.E. (2006). Random number generation and Monte Carlo methods. Springer.

See Also

Hypergeometric

Examples

# Generating 10 random draws from multivariate hypergeometric
# distribution parametrized using a vector

rmvhyper(10, c(10, 12, 5, 8, 11), 33)

Multinomial distribution

Description

Probability mass function and random generation for the multinomial distribution.

Usage

dmnom(x, size, prob, log = FALSE)

rmnom(n, size, prob)

Arguments

x

kk-column matrix of quantiles.
size

numeric vector; number of trials (zero or more).
prob

kk-column numeric matrix; probability of success on each trial.
log

logical; if TRUE, probabilities p are given as log(p).
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

f(x)=n!∏i=1kxi∏i=1kpixif(x) = \frac{n!}{\prod_{i=1}^k x_i} \prod_{i=1}^k p_i^{x_i}

References

Gentle, J.E. (2006). Random number generation and Monte Carlo methods. Springer.

See Also

Binomial, Multinomial

Examples

# Generating 10 random draws from multinomial distribution
# parametrized using a vector

(x <- rmnom(10, 3, c(1/3, 1/3, 1/3)))

# Results are consistent with dmultinom() from stats:

all.equal(dmultinom(x[1,], 3, c(1/3, 1/3, 1/3)),
          dmnom(x[1, , drop = FALSE], 3, c(1/3, 1/3, 1/3)))

Negative hypergeometric distribution

Description

Probability mass function, distribution function, quantile function and random generation for the negative hypergeometric distribution.

Usage

dnhyper(x, n, m, r, log = FALSE)

pnhyper(q, n, m, r, lower.tail = TRUE, log.p = FALSE)

qnhyper(p, n, m, r, lower.tail = TRUE, log.p = FALSE)

rnhyper(nn, n, m, r)

Arguments

x, q

vector of quantiles representing the number of balls drawn without replacement from an urn which contains both black and white balls.
n

the number of black balls in the urn.
m

the number of white balls in the urn.
r

the number of white balls that needs to be drawn for the sampling to be stopped.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
nn

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Negative hypergeometric distribution describes number of balls xx observed until drawing without replacement to obtain rr white balls from the urn containing mm white balls and nn black balls, and is defined as

f(x)=(x−1r−1)(m+n−xm−r)(m+nn)f(x) = \frac{{x-1 \choose r-1}{m+n-x \choose m-r}}{{m+n \choose n}}

The algorithm used for calculating probability mass function, cumulative distribution function and quantile function is based on Fortran program NHYPERG created by Berry and Mielke (1996, 1998). Random generation is done by inverse transform sampling.

References

Berry, K. J., & Mielke, P. W. (1998). The negative hypergeometric probability distribution: Sampling without replacement from a finite population. Perceptual and motor skills, 86(1), 207-210. https://journals.sagepub.com/doi/10.2466/pms.1998.86.1.207

Berry, K. J., & Mielke, P. W. (1996). Exact confidence limits for population proportions based on the negative hypergeometric probability distribution. Perceptual and motor skills, 83(3 suppl), 1216-1218. https://journals.sagepub.com/doi/10.2466/pms.1996.83.3f.1216

Schuster, E. F., & Sype, W. R. (1987). On the negative hypergeometric distribution. International Journal of Mathematical Education in Science and Technology, 18(3), 453-459.

Chae, K. C. (1993). Presenting the negative hypergeometric distribution to the introductory statistics courses. International Journal of Mathematical Education in Science and Technology, 24(4), 523-526.

Jones, S.N. (2013). A Gaming Application of the Negative Hypergeometric Distribution. UNLV Theses, Dissertations, Professional Papers, and Capstones. Paper 1846. https://digitalscholarship.unlv.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=2847&context=thesesdissertations

See Also

Hypergeometric

Examples

x <- rnhyper(1e5, 60, 35, 15)
xx <- 15:95
plot(prop.table(table(x)))
lines(xx, dnhyper(xx, 60, 35, 15), col = "red")
hist(pnhyper(x, 60, 35, 15))

xx <- seq(0, 100, by = 0.01)
plot(ecdf(x))
lines(xx, pnhyper(xx, 60, 35, 15), col = "red", lwd = 2)

Mixture of normal distributions

Description

Density, distribution function and random generation for the mixture of normal distributions.

Usage

dmixnorm(x, mean, sd, alpha, log = FALSE)

pmixnorm(q, mean, sd, alpha, lower.tail = TRUE, log.p = FALSE)

rmixnorm(n, mean, sd, alpha)

Arguments

x, q

vector of quantiles.
mean

matrix (or vector) of means.
sd

matrix (or vector) of standard deviations.
alpha

matrix (or vector) of mixing proportions; mixing proportions need to sum up to 1.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.
p

vector of probabilities.

Details

Probability density function

f(x)=α1f1(x;μ1,σ1)+⋯+αkfk(x;μk,σk)f(x) = \alpha_1 f_1(x; \mu_1, \sigma_1) + \dots + \alpha_k f_k(x; \mu_k, \sigma_k)

Cumulative distribution function

F(x)=α1F1(x;μ1,σ1)+⋯+αkFk(x;μk,σk)F(x) = \alpha_1 F_1(x; \mu_1, \sigma_1) + \dots + \alpha_k F_k(x; \mu_k, \sigma_k)

where ∑iαi=1\sum_i \alpha_i = 1.

Examples

x <- rmixnorm(1e5, c(0.5, 3, 6), c(3, 1, 1), c(1/3, 1/3, 1/3))
hist(x, 100, freq = FALSE)
curve(dmixnorm(x, c(0.5, 3, 6), c(3, 1, 1), c(1/3, 1/3, 1/3)),
      -20, 20, n = 500, col = "red", add = TRUE)
hist(pmixnorm(x, c(0.5, 3, 6), c(3, 1, 1), c(1/3, 1/3, 1/3)))
plot(ecdf(x))
curve(pmixnorm(x, c(0.5, 3, 6), c(3, 1, 1), c(1/3, 1/3, 1/3)),
      -20, 20, n = 500, col = "red", lwd = 2, add = TRUE)

Non-standard beta distribution

Description

Non-standard form of beta distribution with lower and upper bounds denoted as min and max. By default min=0 and max=1 what leads to standard beta distribution.

Usage

dnsbeta(x, shape1, shape2, min = 0, max = 1, log = FALSE)

pnsbeta(q, shape1, shape2, min = 0, max = 1, lower.tail = TRUE, log.p = FALSE)

qnsbeta(p, shape1, shape2, min = 0, max = 1, lower.tail = TRUE, log.p = FALSE)

rnsbeta(n, shape1, shape2, min = 0, max = 1)

Arguments

x, q

vector of quantiles.
shape1, shape2

non-negative parameters of the Beta distribution.
min, max

lower and upper bounds.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \leq x], otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

See Also

Beta

Examples

x <- rnsbeta(1e5, 5, 13, -4, 8)
hist(x, 100, freq = FALSE)
curve(dnsbeta(x, 5, 13, -4, 8), -4, 6, col = "red", add = TRUE) 
hist(pnsbeta(x, 5, 13, -4, 8))
plot(ecdf(x))
curve(pnsbeta(x, 5, 13, -4, 8), -4, 6, col = "red", lwd = 2, add = TRUE)

Pareto distribution

Description

Density, distribution function, quantile function and random generation for the Pareto distribution.

Usage

dpareto(x, a = 1, b = 1, log = FALSE)

ppareto(q, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)

qpareto(p, a = 1, b = 1, lower.tail = TRUE, log.p = FALSE)

rpareto(n, a = 1, b = 1)

Arguments

x, q

vector of quantiles.
a, b

positive valued scale and location parameters.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=abaxa+1f(x) = \frac{ab^a}{x^{a+1}}

Cumulative distribution function

F(x)=1−(bx)aF(x) = 1 - \left(\frac{b}{x}\right)^a

Quantile function

F−1(p)=b(1−p)1−aF^{-1}(p) = \frac{b}{(1-p)^{1-a}}

References

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC

Examples

x <- rpareto(1e5, 5, 16)
hist(x, 100, freq = FALSE)
curve(dpareto(x, 5, 16), 0, 200, col = "red", add = TRUE)
hist(ppareto(x, 5, 16))
plot(ecdf(x))
curve(ppareto(x, 5, 16), 0, 200, col = "red", lwd = 2, add = TRUE)

Mixture of Poisson distributions

Description

Density, distribution function and random generation for the mixture of Poisson distributions.

Usage

dmixpois(x, lambda, alpha, log = FALSE)

pmixpois(q, lambda, alpha, lower.tail = TRUE, log.p = FALSE)

rmixpois(n, lambda, alpha)

Arguments

x, q

vector of quantiles.
lambda

matrix (or vector) of (non-negative) means.
alpha

matrix (or vector) of mixing proportions; mixing proportions need to sum up to 1.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.
p

vector of probabilities.

Details

Probability density function

f(x)=α1f1(x;λ1)+⋯+αkfk(x;λk)f(x) = \alpha_1 f_1(x; \lambda_1) + \dots + \alpha_k f_k(x; \lambda_k)

Cumulative distribution function

F(x)=α1F1(x;λ1)+⋯+αkFk(x;λk)F(x) = \alpha_1 F_1(x; \lambda_1) + \dots + \alpha_k F_k(x; \lambda_k)

where ∑iαi=1\sum_i \alpha_i = 1.

Examples

x <- rmixpois(1e5, c(5, 12, 19), c(1/3, 1/3, 1/3))
xx <- seq(-1, 50)
plot(prop.table(table(x)))
lines(xx, dmixpois(xx, c(5, 12, 19), c(1/3, 1/3, 1/3)), col = "red")
hist(pmixpois(x, c(5, 12, 19), c(1/3, 1/3, 1/3)))

xx <- seq(0, 50, by = 0.01)
plot(ecdf(x))
lines(xx, pmixpois(xx, c(5, 12, 19), c(1/3, 1/3, 1/3)), col = "red", lwd = 2)

Power distribution

Description

Density, distribution function, quantile function and random generation for the power distribution.

Usage

dpower(x, alpha, beta, log = FALSE)

ppower(q, alpha, beta, lower.tail = TRUE, log.p = FALSE)

qpower(p, alpha, beta, lower.tail = TRUE, log.p = FALSE)

rpower(n, alpha, beta)

Arguments

x, q

vector of quantiles.
alpha, beta

parameters.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=βxβ−1αβf(x) = \frac{\beta x^{\beta-1}}{\alpha^\beta}

Cumulative distribution function

F(x)=xβαβF(x) = \frac{x^\beta}{\alpha^\beta}

Quantile function

F−1(p)=αp1/βF^{-1}(p) = \alpha p^{1/\beta}

Examples

x <- rpower(1e5, 5, 16)
hist(x, 100, freq = FALSE)
curve(dpower(x, 5, 16), 2, 6, col = "red", add = TRUE, n = 5000)
hist(ppower(x, 5, 16))
plot(ecdf(x))
curve(ppower(x, 5, 16), 2, 6, col = "red", lwd = 2, add = TRUE)

Beta distribution of proportions

Description

Probability mass function, distribution function and random generation for the reparametrized beta distribution.

Usage

dprop(x, size, mean, prior = 0, log = FALSE)

pprop(q, size, mean, prior = 0, lower.tail = TRUE, log.p = FALSE)

qprop(p, size, mean, prior = 0, lower.tail = TRUE, log.p = FALSE)

rprop(n, size, mean, prior = 0)

Arguments

x, q

vector of quantiles.
size

non-negative real number; precision or number of binomial trials.
mean

mean proportion or probability of success on each trial; 0 < mean < 1.
prior

(see below) with prior = 0 (default) the distribution corresponds to re-parametrized beta distribution used in beta regression. This parameter needs to be non-negative.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Beta can be understood as a distribution of x=k/ϕx = k/\phi proportions in ϕ\phi trials where the average proportion is denoted as μ\mu, so it's parameters become α=ϕμ\alpha = \phi\mu and β=ϕ(1−μ)\beta = \phi(1-\mu) and it's density function becomes

f(x)=xϕμ+π−1(1−x)ϕ(1−μ)+π−1B(ϕμ+π,ϕ(1−μ)+π)f(x) = \frac{x^{\phi\mu+\pi-1} (1-x)^{\phi(1-\mu)+\pi-1}}{\mathrm{B}(\phi\mu+\pi, \phi(1-\mu)+\pi)}

where π\pi is a prior parameter, so the distribution is a posterior distribution after observing ϕμ\phi\mu successes and ϕ(1−μ)\phi(1-\mu) failures in ϕ\phi trials with binomial likelihood and symmetric Beta(π,π)\mathrm{Beta}(\pi, \pi) prior for probability of success. Parameter value π=1\pi = 1 corresponds to uniform prior; π=1/2\pi = 1/2 corresponds to Jeffreys prior; π=0\pi = 0 corresponds to "uninformative" Haldane prior, this is also the re-parametrized distribution used in beta regression. With π=0\pi = 0 the distribution can be understood as a continuous analog to binomial distribution dealing with proportions rather then counts. Alternatively ϕ\phi may be understood as precision parameter (as in beta regression).

Notice that in pre-1.8.4 versions of this package, prior was not settable and by default fixed to one, instead of zero. To obtain the same results as in the previous versions, use prior = 1 in each of the functions.

References

Ferrari, S., & Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799-815.

Smithson, M., & Verkuilen, J. (2006). A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychological Methods, 11(1), 54-71.

See Also

beta, binomial

Examples

x <- rprop(1e5, 100, 0.33)
hist(x, 100, freq = FALSE)
curve(dprop(x, 100, 0.33), 0, 1, col = "red", add = TRUE)
hist(pprop(x, 100, 0.33))
plot(ecdf(x))
curve(pprop(x, 100, 0.33), 0, 1, col = "red", lwd = 2, add = TRUE)

n <- 500
p <- 0.23
k <- rbinom(1e5, n, p)
hist(k/n, freq = FALSE, 100)
curve(dprop(x, n, p), 0, 1, col = "red", add = TRUE, n = 500)

Random generation from Rademacher distribution

Description

Random generation for the Rademacher distribution (values -1 and +1 with equal probability).

Usage

rsign(n)

Arguments

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Rayleigh distribution

Description

Density, distribution function, quantile function and random generation for the Rayleigh distribution.

Usage

drayleigh(x, sigma = 1, log = FALSE)

prayleigh(q, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qrayleigh(p, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rrayleigh(n, sigma = 1)

Arguments

x, q

vector of quantiles.
sigma

positive valued parameter.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=xσ2exp⁡(−x22σ2)f(x) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2\sigma^2}\right)

Cumulative distribution function

F(x)=1−exp⁡(−x22σ2)F(x) = 1 - \exp\left(-\frac{x^2}{2\sigma^2}\right)

Quantile function

F−1(p)=−2σ2log⁡(1−p)F^{-1}(p) = \sqrt{-2\sigma^2 \log(1-p)}

References

Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC.

Forbes, C., Evans, M. Hastings, N., & Peacock, B. (2011). Statistical Distributions. John Wiley & Sons.

Examples

x <- rrayleigh(1e5, 13)
hist(x, 100, freq = FALSE)
curve(drayleigh(x, 13), 0, 60, col = "red", add = TRUE)
hist(prayleigh(x, 13)) 
plot(ecdf(x))
curve(prayleigh(x, 13), 0, 60, col = "red", lwd = 2, add = TRUE)

Shifted Gompertz distribution

Description

Density, distribution function, and random generation for the shifted Gompertz distribution.

Usage

dsgomp(x, b, eta, log = FALSE)

psgomp(q, b, eta, lower.tail = TRUE, log.p = FALSE)

rsgomp(n, b, eta)

Arguments

x, q

vector of quantiles.
b, eta

positive valued scale and shape parameters; both need to be positive.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If XX follows exponential distribution parametrized by scale bb and YY follows reparametrized Gumbel distribution with cumulative distribution function F(x)=exp⁡(−ηe−bx)F(x) = \exp(-\eta e^{-bx}) parametrized by scale bb and shape η\eta, then max⁡(X,Y)\max(X,Y) follows shifted Gompertz distribution parametrized by scale b>0b>0 and shape η>0\eta>0. The above relation is used by rsgomp function for random generation from shifted Gompertz distribution.

Probability density function

f(x)=be−bxexp⁡(−ηe−bx)[1+η(1−e−bx)]f(x) = b e^{-bx} \exp(-\eta e^{-bx}) \left[1 + \eta(1 - e^{-bx})\right]

Cumulative distribution function

F(x)=(1−e−bx)exp⁡(−ηe−bx)F(x) = (1-e^{-bx}) \exp(-\eta e^{-bx})

References

Bemmaor, A.C. (1994). Modeling the Diffusion of New Durable Goods: Word-of-Mouth Effect Versus Consumer Heterogeneity. [In:] G. Laurent, G.L. Lilien & B. Pras. Research Traditions in Marketing. Boston: Kluwer Academic Publishers. pp. 201-223.

Jimenez, T.F. and Jodra, P. (2009). A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution. Communications in Statistics - Theory and Methods, 38(1), 78-89.

Jimenez T.F. (2014). Estimation of the Parameters of the Shifted Gompertz Distribution, Using Least Squares, Maximum Likelihood and Moments Methods. Journal of Computational and Applied Mathematics, 255(1), 867-877.

Examples

x <- rsgomp(1e5, 0.4, 1)
hist(x, 50, freq = FALSE)
curve(dsgomp(x, 0.4, 1), 0, 30, col = "red", add = TRUE)
hist(psgomp(x, 0.4, 1))
plot(ecdf(x))
curve(psgomp(x, 0.4, 1), 0, 30, col = "red", lwd = 2, add = TRUE)

Skellam distribution

Description

Probability mass function and random generation for the Skellam distribution.

Usage

dskellam(x, mu1, mu2, log = FALSE)

rskellam(n, mu1, mu2)

Arguments

x

vector of quantiles.
mu1, mu2

positive valued parameters.
log

logical; if TRUE, probabilities p are given as log(p).
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If XX and YY follow Poisson distributions with means μ1\mu_1 and μ2\mu_2, than X−YX-Y follows Skellam distribution parametrized by μ1\mu_1 and μ2\mu_2.

Probability mass function

f(x)=e−(μ1 ⁣+ ⁣μ2)(μ1μ2)k/2 ⁣ ⁣Ik(2μ1μ2)f(x) = e^{-(\mu_1\!+\!\mu_2)} \left(\frac{\mu_1}{\mu_2}\right)^{k/2}\!\!I_{k}(2\sqrt{\mu_1\mu_2})

References

Karlis, D., & Ntzoufras, I. (2006). Bayesian analysis of the differences of count data. Statistics in medicine, 25(11), 1885-1905.

Skellam, J.G. (1946). The frequency distribution of the difference between two Poisson variates belonging to different populations. Journal of the Royal Statistical Society, series A, 109(3), 26.

Examples

x <- rskellam(1e5, 5, 13)
xx <- -40:40
plot(prop.table(table(x)), type = "h")
lines(xx, dskellam(xx, 5, 13), col = "red")

Slash distribution

Description

Probability mass function, distribution function and random generation for slash distribution.

Usage

dslash(x, mu = 0, sigma = 1, log = FALSE)

pslash(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rslash(n, mu = 0, sigma = 1)

Arguments

x, q

vector of quantiles.
mu

vector of locations
sigma

vector of positive valued scale parameters.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

If Z∼Normal(0,1)Z \sim \mathrm{Normal}(0, 1) and U∼Uniform(0,1)U \sim \mathrm{Uniform}(0, 1), then Z/UZ/U follows slash distribution.

Probability density function

f(x)={ϕ(0)−ϕ(x)x2x≠0122πx=0f(x) = \left\{\begin{array}{ll} \frac{\phi(0) - \phi(x)}{x^2} & x \ne 0 \\ \frac{1}{2\sqrt{2\pi}} & x = 0 \end{array}\right.

Cumulative distribution function

F(x)={Φ(x)−ϕ(0)−ϕ(x)xx≠012x=0F(x) = \left\{\begin{array}{ll} \Phi(x) - \frac{\phi(0)-\phi(x)}{x} & x \neq 0 \\ \frac{1}{2} & x = 0 \end{array}\right.

Examples

x <- rslash(1e5, 5, 3)
hist(x, 1e5, freq = FALSE, xlim = c(-100, 100))
curve(dslash(x, 5, 3), -100, 100, col = "red", n = 500, add = TRUE)
hist(pslash(x, 5, 3))
plot(ecdf(x), xlim = c(-100, 100))
curve(pslash(x, 5, 3), -100, 100, col = "red", lwd = 2, n = 500, add = TRUE)

Triangular distribution

Description

Density, distribution function, quantile function and random generation for the triangular distribution.

Usage

dtriang(x, a = -1, b = 1, c = (a + b)/2, log = FALSE)

ptriang(q, a = -1, b = 1, c = (a + b)/2, lower.tail = TRUE, log.p = FALSE)

qtriang(p, a = -1, b = 1, c = (a + b)/2, lower.tail = TRUE, log.p = FALSE)

rtriang(n, a = -1, b = 1, c = (a + b)/2)

Arguments

x, q

vector of quantiles.
a, b, c

minimum, maximum and mode of the distribution.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)={2(x−a)(b−a)(c−a)x<c2b−ax=c2(b−x)(b−a)(b−c)x>cf(x) = \left\{\begin{array}{ll} \frac{2(x-a)}{(b-a)(c-a)} & x < c \\ \frac{2}{b-a} & x = c \\ \frac{2(b-x)}{(b-a)(b-c)} & x > c \end{array}\right.

Cumulative distribution function

F(x)={(x−a)2(b−a)(c−a)x≤c1−(b−x)2(b−a)(b−c)x>cF(x) = \left\{\begin{array}{ll} \frac{(x-a)^2}{(b-a)(c-a)} & x \leq c \\ 1 - \frac{(b-x)^2}{(b-a)(b-c)} & x > c \end{array}\right.

Quantile function

F−1(p)={a+p×(b−a)(c−a)p≤c−ab−ab−(1−p)(b−a)(b−c)p>c−ab−aF^{-1}(p) = \left\{\begin{array}{ll} a + \sqrt{p \times (b-a)(c-a)} & p \leq \frac{c-a}{b-a} \\ b - \sqrt{(1-p)(b-a)(b-c)} & p > \frac{c-a}{b-a} \end{array}\right.

For random generation MINMAX method described by Stein and Keblis (2009) is used.

References

Forbes, C., Evans, M. Hastings, N., & Peacock, B. (2011). Statistical Distributions. John Wiley & Sons.

Stein, W. E., & Keblis, M. F. (2009). A new method to simulate the triangular distribution. Mathematical and computer modelling, 49(5), 1143-1147.

Examples

x <- rtriang(1e5, 5, 7, 6)
hist(x, 100, freq = FALSE)
curve(dtriang(x, 5, 7, 6), 3, 10, n = 500, col = "red", add = TRUE)
hist(ptriang(x, 5, 7, 6))
plot(ecdf(x))
curve(ptriang(x, 5, 7, 6), 3, 10, n = 500, col = "red", lwd = 2, add = TRUE)

Truncated binomial distribution

Description

Density, distribution function, quantile function and random generation for the truncated binomial distribution.

Usage

dtbinom(x, size, prob, a = -Inf, b = Inf, log = FALSE)

ptbinom(q, size, prob, a = -Inf, b = Inf, lower.tail = TRUE, log.p = FALSE)

qtbinom(p, size, prob, a = -Inf, b = Inf, lower.tail = TRUE, log.p = FALSE)

rtbinom(n, size, prob, a = -Inf, b = Inf)

Arguments

x, q

vector of quantiles.
size

number of trials (zero or more).
prob

probability of success on each trial.
a, b

lower and upper truncation points (a < x <= b).
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Examples

x <- rtbinom(1e5, 100, 0.83, 76, 86)
xx <- seq(0, 100)
plot(prop.table(table(x)))
lines(xx, dtbinom(xx, 100, 0.83, 76, 86), col = "red")
hist(ptbinom(x, 100, 0.83, 76, 86))

xx <- seq(0, 100, by = 0.01)
plot(ecdf(x))
lines(xx, ptbinom(xx, 100, 0.83, 76, 86), col = "red", lwd = 2)
uu <- seq(0, 1, by = 0.001)
lines(qtbinom(uu, 100, 0.83, 76, 86), uu, col = "blue", lty = 2)

Truncated normal distribution

Description

Density, distribution function, quantile function and random generation for the truncated normal distribution.

Usage

dtnorm(x, mean = 0, sd = 1, a = -Inf, b = Inf, log = FALSE)

ptnorm(
  q,
  mean = 0,
  sd = 1,
  a = -Inf,
  b = Inf,
  lower.tail = TRUE,
  log.p = FALSE
)

qtnorm(
  p,
  mean = 0,
  sd = 1,
  a = -Inf,
  b = Inf,
  lower.tail = TRUE,
  log.p = FALSE
)

rtnorm(n, mean = 0, sd = 1, a = -Inf, b = Inf)

Arguments

x, q

vector of quantiles.
mean, sd

location and scale parameters. Scale must be positive.
a, b

lower and upper truncation points (a < x <= b, with a = -Inf and b = Inf by default).
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)=ϕ(x−μσ)Φ(b−μσ)−Φ(a−μσ)f(x) = \frac{\phi(\frac{x-\mu}{\sigma})} {\Phi(\frac{b-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma})}

Cumulative distribution function

F(x)=Φ(x−μσ)−Φ(a−μσ)Φ(b−μσ)−Φ(a−μσ)F(x) = \frac{\Phi(\frac{x-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma})} {\Phi(\frac{b-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma})}

Quantile function

F−1(p)=Φ−1(Φ(a−μσ)+p×[Φ(b−μσ)−Φ(a−μσ)])F^{-1}(p) = \Phi^{-1}\left(\Phi\left(\frac{a-\mu}{\sigma}\right) + p \times \left[\Phi\left(\frac{b-\mu}{\sigma}\right) - \Phi\left(\frac{a-\mu}{\sigma}\right)\right]\right)

For random generation algorithm described by Robert (1995) is used.

References

Robert, C.P. (1995). Simulation of truncated normal variables. Statistics and Computing 5(2): 121-125. https://arxiv.org/abs/0907.4010

Burkardt, J. (17 October 2014). The Truncated Normal Distribution. Florida State University. https://people.sc.fsu.edu/~jburkardt/presentations/truncated_normal.pdf

Examples

x <- rtnorm(1e5, 5, 3, b = 7)
hist(x, 100, freq = FALSE)
curve(dtnorm(x, 5, 3, b = 7), -8, 8, col = "red", add = TRUE)
hist(ptnorm(x, 5, 3, b = 7))
plot(ecdf(x))
curve(ptnorm(x, 5, 3, b = 7), -8, 8, col = "red", lwd = 2, add = TRUE)

R <- 1e5
partmp <- par(mfrow = c(2,4), mar = c(2,2,2,2))

hist(rtnorm(R), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x), -5, 5, col = "red", add = TRUE)

hist(rtnorm(R, a = 0), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, a = 0), -1, 5, col = "red", add = TRUE)

hist(rtnorm(R, b = 0), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, b = 0), -5, 5, col = "red", add = TRUE)

hist(rtnorm(R, a = 0, b = 1), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, a = 0, b = 1), -1, 2, col = "red", add = TRUE)

hist(rtnorm(R, a = -1, b = 0), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, a = -1, b = 0), -2, 2, col = "red", add = TRUE)

hist(rtnorm(R, mean = -6, a = 0), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, mean = -6, a = 0), -2, 1, col = "red", add = TRUE)

hist(rtnorm(R, mean = 8, b = 0), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, mean = 8, b = 0), -2, 1, col = "red", add = TRUE)

hist(rtnorm(R, a = 3, b = 5), freq= FALSE, main = "", xlab = "", ylab = "")
curve(dtnorm(x, a = 3, b = 5), 2, 5, col = "red", add = TRUE)

par(partmp)

Truncated Poisson distribution

Description

Density, distribution function, quantile function and random generation for the truncated Poisson distribution.

Usage

dtpois(x, lambda, a = -Inf, b = Inf, log = FALSE)

ptpois(q, lambda, a = -Inf, b = Inf, lower.tail = TRUE, log.p = FALSE)

qtpois(p, lambda, a = -Inf, b = Inf, lower.tail = TRUE, log.p = FALSE)

rtpois(n, lambda, a = -Inf, b = Inf)

Arguments

x, q

vector of quantiles.
lambda

vector of (non-negative) means.
a, b

lower and upper truncation points (a < x <= b).
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

References

Plackett, R.L. (1953). The truncated Poisson distribution. Biometrics, 9(4), 485-488.

Singh, J. (1978). A characterization of positive Poisson distribution and its statistical application. SIAM Journal on Applied Mathematics, 34(3), 545-548.

Dalgaard, P. (May 1, 2005). [R] simulate zero-truncated Poisson distribution. R-help mailing list. https://stat.ethz.ch/pipermail/r-help/2005-May/070680.html

Examples

x <- rtpois(1e5, 14, 16)
xx <- seq(-1, 50)
plot(prop.table(table(x)))
lines(xx, dtpois(xx, 14, 16), col = "red")
hist(ptpois(x, 14, 16))

xx <- seq(0, 50, by = 0.01)
plot(ecdf(x))
lines(xx, ptpois(xx, 14, 16), col = "red", lwd = 2)

uu <- seq(0, 1, by = 0.001)
lines(qtpois(uu, 14, 16), uu, col = "blue", lty = 2)

# Zero-truncated Poisson

x <- rtpois(1e5, 5, 0)
xx <- seq(-1, 50)
plot(prop.table(table(x)))
lines(xx, dtpois(xx, 5, 0), col = "red")
hist(ptpois(x, 5, 0))

xx <- seq(0, 50, by = 0.01)
plot(ecdf(x))
lines(xx, ptpois(xx, 5, 0), col = "red", lwd = 2)
lines(qtpois(uu, 5, 0), uu, col = "blue", lty = 2)

Tukey lambda distribution

Description

Quantile function, and random generation for the Tukey lambda distribution.

Usage

qtlambda(p, lambda, lower.tail = TRUE, log.p = FALSE)

rtlambda(n, lambda)

Arguments

p

vector of probabilities.
lambda

shape parameter.
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
log.p

logical; if TRUE, probabilities p are given as log(p).
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Tukey lambda distribution is a continuous probability distribution defined in terms of its quantile function. It is typically used to identify other distributions.

Quantile function:

F−1(p)={1λ[pλ−(1−p)λ]λ≠0log⁡(p1−p)λ=0F^{-1}(p) = \left\{\begin{array}{ll} \frac{1}{\lambda} [p^\lambda - (1-p)^\lambda] & \lambda \ne 0 \\ \log(\frac{p}{1-p}) & \lambda = 0 \end{array}\right.

References

Joiner, B.L., & Rosenblatt, J.R. (1971). Some properties of the range in samples from Tukey's symmetric lambda distributions. Journal of the American Statistical Association, 66(334), 394-399.

Hastings Jr, C., Mosteller, F., Tukey, J.W., & Winsor, C.P. (1947). Low moments for small samples: a comparative study of order statistics. The Annals of Mathematical Statistics, 413-426.

Examples

pp = seq(0, 1, by = 0.001)
partmp <- par(mfrow = c(2,3))
plot(qtlambda(pp, -1), pp, type = "l", main = "lambda = -1 (Cauchy)")
plot(qtlambda(pp, 0), pp, type = "l", main = "lambda = 0 (logistic)")
plot(qtlambda(pp, 0.14), pp, type = "l", main = "lambda = 0.14 (normal)")
plot(qtlambda(pp, 0.5), pp, type = "l", main = "lambda = 0.5 (concave)")
plot(qtlambda(pp, 1), pp, type = "l", main = "lambda = 1 (uniform)")
plot(qtlambda(pp, 2), pp, type = "l", main = "lambda = 2 (uniform)")

hist(rtlambda(1e5, -1), freq = FALSE, main = "lambda = -1 (Cauchy)")
hist(rtlambda(1e5, 0), freq = FALSE, main = "lambda = 0 (logistic)")
hist(rtlambda(1e5, 0.14), freq = FALSE, main = "lambda = 0.14 (normal)")
hist(rtlambda(1e5, 0.5), freq = FALSE, main = "lambda = 0.5 (concave)")
hist(rtlambda(1e5, 1), freq = FALSE, main = "lambda = 1 (uniform)")
hist(rtlambda(1e5, 2), freq = FALSE, main = "lambda = 2 (uniform)")
par(partmp)

Wald (inverse Gaussian) distribution

Description

Density, distribution function and random generation for the Wald distribution.

Usage

dwald(x, mu, lambda, log = FALSE)

pwald(q, mu, lambda, lower.tail = TRUE, log.p = FALSE)

rwald(n, mu, lambda)

Arguments

x, q

vector of quantiles.
mu, lambda

location and shape parameters. Scale must be positive.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
n

number of observations. If length(n) > 1, the length is taken to be the number required.
p

vector of probabilities.

Details

Probability density function

f(x)=λ2πx3exp⁡(−λ(x−μ)22μ2x)f(x) = \sqrt{\frac{\lambda}{2\pi x^3}} \exp\left( \frac{-\lambda(x-\mu)^2}{2\mu^2 x} \right)

Cumulative distribution function

F(x)=Φ(λx(xμ−1))+exp⁡(2λμ)Φ(λx(xμ+1))F(x) = \Phi\left(\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu}-1 \right) \right) + \exp\left(\frac{2\lambda}{\mu} \right) \Phi\left(\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu}+1 \right) \right)

Random generation is done using the algorithm described by Michael, Schucany and Haas (1976).

References

Michael, J.R., Schucany, W.R., and Haas, R.W. (1976). Generating Random Variates Using Transformations with Multiple Roots. The American Statistician, 30(2): 88-90.

Examples

x <- rwald(1e5, 5, 16)
hist(x, 100, freq = FALSE)
curve(dwald(x, 5, 16), 0, 50, col = "red", add = TRUE)
hist(pwald(x, 5, 16))
plot(ecdf(x))
curve(pwald(x, 5, 16), 0, 50, col = "red", lwd = 2, add = TRUE)

Zero-inflated binomial distribution

Description

Probability mass function and random generation for the zero-inflated binomial distribution.

Usage

dzib(x, size, prob, pi, log = FALSE)

pzib(q, size, prob, pi, lower.tail = TRUE, log.p = FALSE)

qzib(p, size, prob, pi, lower.tail = TRUE, log.p = FALSE)

rzib(n, size, prob, pi)

Arguments

x, q

vector of quantiles.
size

number of trials (zero or more).
prob

probability of success in each trial. 0 < prob <= 1.
pi

probability of extra zeros.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)={π+(1−π)(1−p)nx=0(1−π)(nx)px(1−p)n−xx>0f(x) = \left\{\begin{array}{ll} \pi + (1 - \pi) (1-p)^n & x = 0 \\ (1 - \pi) {n \choose x} p^x (1-p)^{n-x} & x > 0 \\ \end{array}\right.

See Also

Binomial

Examples

x <- rzib(1e5, 10, 0.6, 0.33)
xx <- -2:20
plot(prop.table(table(x)), type = "h")
lines(xx, dzib(xx, 10, 0.6, 0.33), col = "red")

xx <- seq(0, 20, by = 0.01)
plot(ecdf(x))
lines(xx, pzib(xx, 10, 0.6, 0.33), col = "red")

Zero-inflated negative binomial distribution

Description

Probability mass function and random generation for the zero-inflated negative binomial distribution.

Usage

dzinb(x, size, prob, pi, log = FALSE)

pzinb(q, size, prob, pi, lower.tail = TRUE, log.p = FALSE)

qzinb(p, size, prob, pi, lower.tail = TRUE, log.p = FALSE)

rzinb(n, size, prob, pi)

Arguments

x, q

vector of quantiles.
size

target for number of successful trials, or dispersion parameter (the shape parameter of the gamma mixing distribution). Must be strictly positive, need not be integer.
prob

probability of success in each trial. 0 < prob <= 1.
pi

probability of extra zeros.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)={π+(1−π)prx=0(1−π)(x+r−1x)pr(1−p)xx>0f(x) = \left\{\begin{array}{ll} \pi + (1 - \pi) p^r & x = 0 \\ (1 - \pi) {x+r-1 \choose x} p^r (1-p)^x & x > 0 \\ \end{array}\right.

See Also

NegBinomial

Examples

x <- rzinb(1e5, 100, 0.6, 0.33)
xx <- -2:200
plot(prop.table(table(x)), type = "h")
lines(xx, dzinb(xx, 100, 0.6, 0.33), col = "red")

xx <- seq(0, 200, by = 0.01)
plot(ecdf(x))
lines(xx, pzinb(xx, 100, 0.6, 0.33), col = "red")

Zero-inflated Poisson distribution

Description

Probability mass function and random generation for the zero-inflated Poisson distribution.

Usage

dzip(x, lambda, pi, log = FALSE)

pzip(q, lambda, pi, lower.tail = TRUE, log.p = FALSE)

qzip(p, lambda, pi, lower.tail = TRUE, log.p = FALSE)

rzip(n, lambda, pi)

Arguments

x, q

vector of quantiles.
lambda

vector of (non-negative) means.
pi

probability of extra zeros.
log, log.p

logical; if TRUE, probabilities p are given as log(p).
lower.tail

logical; if TRUE (default), probabilities are P[X≤x]P[X \le x] otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)={π+(1−π)e−λx=0(1−π)λxe−λx!x>0f(x) = \left\{\begin{array}{ll} \pi + (1 - \pi) e^{-\lambda} & x = 0 \\ (1 - \pi) \frac{\lambda^{x} e^{-\lambda}} {x!} & x > 0 \\ \end{array}\right.

See Also

Poisson

Examples

x <- rzip(1e5, 6, 0.33)
xx <- -2:20
plot(prop.table(table(x)), type = "h")
lines(xx, dzip(xx, 6, 0.33), col = "red")

xx <- seq(0, 20, by = 0.01)
plot(ecdf(x))
lines(xx, pzip(xx, 6, 0.33), col = "red")