extraDistr: 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.

Density, distribution function, quantile function and random generation for a number of univariate and multivariate distributions.

Details

This package follows naming convention that is consistent with base R, where density (or probability mass) functions, distribution functions, quantile functions and random generation functions names are followed by d*, p*, q*, and r* prefixes.

Behaviour of the functions is consistent with base R, where for not valid parameters values NaN's are returned, while for values beyond function support 0's are returned (e.g. for non-integers in discrete distributions, or for negative values in functions with non-negative support).

All the functions vectorized and coded in C++ using Rcpp.

See Also

Useful links:

• https://github.com/twolodzko/extraDistr

• Report bugs at https://github.com/twolodzko/extraDistr/issues

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

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

Details

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

Probability mass function

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 \Gamma(x) = (x - 1)!, and \mathrm{B}(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} , and that {n \choose k} = \prod_{i=1}^k \frac{n+1-i}{i} . This enables re-writing probability mass function as

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 x to x+1 easy using the properties of factorials

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) = \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

Details

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

Probability mass function

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 \Gamma(x) = (x - 1)! and \mathrm{B}(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} . This enables re-writing probability mass function as

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 x to x+1 easy using the properties of factorials

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) = \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

Details

If X \sim \mathrm{Beta}(\alpha, \beta), then \frac{X}{1-X} \sim \mathrm{BetaPrime}(\alpha, \beta).

Probability density function

f(x) = \frac{(x/\sigma)^{\alpha-1} (1+x/\sigma)^{-\alpha -\beta}}{\mathrm{B}(\alpha,\beta)\sigma}

Cumulative distribution function

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

Details

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

Probability density function

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) = \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

Details

Probability density function

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) = \Phi \left(\frac{1}{\alpha}\left( \sqrt{\frac{x-\mu}{\beta}} - \sqrt{\frac{\beta}{x-\mu}} \right) \right)

Quantile function

F^{-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

Details

Probability density function

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

Details

Probability mass function

f(x) = \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

Details

Probability mass function

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

Cumulative distribution function

\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 \alpha_1,\dots,\alpha_m without leaving the log space by employing the Gumbel-max trick (Maddison, Tarlow and Minka, 2014). If g_1,\dots,g_m are samples from Gumbel distribution with cumulative distribution function F(g) = \exp(-\exp(-g)), then k = \mathrm{arg\,max}_i \{g_i + \alpha_i\} is a draw from categorical distribution parametrized by vector of probabilities p_1,\dots,p_m, such that 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-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

Details

If (p_1,\dots,p_k) \sim \mathrm{Dirichlet}(\alpha_1,\dots,\alpha_k) and (x_1,\dots,x_k) \sim \mathrm{Multinomial}(n, p_1,\dots,p_k), then (x_1,\dots,x_k) \sim \mathrm{DirichletMultinomial(n, \alpha_1,\dots,\alpha_k)}.

Probability density function

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

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

Details

Probability density function

f(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)

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

Details

Probability mass function of discrete gamma distribution f_Y(y) is defined by discretization of continuous gamma distribution f_Y(y) = S_X(y) - S_X(y+1) where S_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

Details

If U \sim \mathrm{Geometric}(1-p) and V \sim \mathrm{Geometric}(1-p), then 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) = \frac{1-p}{1+p} p^{|x-\mu|}

Cumulative distribution function

F(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

Details

Probability mass function

f(x) = \Phi\left(\frac{x-\mu+1}{\sigma}\right) - \Phi\left(\frac{x-\mu}{\sigma}\right)

Cumulative distribution function

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

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

Details

Probability mass function

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

Cumulative distribution function

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

Quantile function

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

Details

Probability density function

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\left(-\left(\frac{x-\mu}{\sigma}\right)^{-\lambda}\right)

Quantile function

F^{-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)

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

Details

Probability density function

f(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) = \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) = \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)

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

Details

Probability density function

f(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) = \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) = \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)

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

Details

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

Probability mass function

f(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 \Gamma(x) = (x - 1)!. This enables re-writing probability mass function as

f(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 x to x+1 easy using the properties of factorials

f(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) = \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)

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

Details

Probability density function

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

Cumulative distribution function

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

Quantile function

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)

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

Details

Probability density function

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\left(-\exp\left(-\frac{x-\mu}{\sigma}\right)\right)

Quantile function

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

Details

If X follows Cauchy centered at 0 and parametrized by scale \sigma, then |X| follows half-Cauchy distribution parametrized by scale \sigma. Half-Cauchy distribution is a special case of half-t distribution with \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

Details

If X follows normal distribution centered at 0 and parametrized by scale \sigma, then |X| follows half-normal distribution parametrized by scale \sigma. Half-t distribution with \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

Details

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

Details

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

f(x) = \frac{1}{2 \sqrt{2\pi} \left( \Phi(k) + \phi(k)/k - \frac{1}{2} \right)} e^{-\rho_k(x)}

where

\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

Details

If X follows \chi^2 (\nu) distribution, then 1/X follows inverse chi-squared distribution parametrized by \nu. Inverse chi-squared distribution is a special case of inverse gamma distribution with parameters \alpha=\frac{\nu}{2} and \beta=\frac{1}{2}; or \alpha=\frac{\nu}{2} and \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

Details

Probability mass function

f(x) = \frac{\beta^\alpha x^{-\alpha-1} \exp(-\frac{\beta}{x})}{\Gamma(\alpha)}

Cumulative distribution function

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

Details

Probability density function

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

Cumulative distribution function

F(x) = 1-(1-x^a)^b

Quantile function

F^{-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

Details

Probability density function

f(x) = \frac{1}{2\sigma} \exp\left(-\left|\frac{x-\mu}{\sigma}\right|\right)

Cumulative distribution function

F(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) = \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 \nu, is parametrized using additional parameters \mu for location and \sigma for scale (\mu = 0 and \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

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

Details

Probability mass function

f(x) = \frac{-1}{\log(1-\theta)} \frac{\theta^x}{x}

Cumulative distribution function

F(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

Details

Probability density function

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

Cumulative distribution function

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

Quantile function

F^{-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

Details

Probability mass function

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 k out of N marbles in m colors, where each of the colors appears n_i times. Where k=\sum_{i=1}^m x_i, N=\sum_{i=1}^m n_i and k \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

Details

Probability mass function

f(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)))

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

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)

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

Details

Negative hypergeometric distribution describes number of balls x observed until drawing without replacement to obtain r white balls from the urn containing m white balls and n black balls, and is defined as

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

Details

Probability density function

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) = \alpha_1 F_1(x; \mu_1, \sigma_1) + \dots + \alpha_k F_k(x; \mu_k, \sigma_k)

where \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)

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

Details

Probability density function

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

Cumulative distribution function

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

Quantile function

F^{-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

Details

Probability density function

f(x) = \alpha_1 f_1(x; \lambda_1) + \dots + \alpha_k f_k(x; \lambda_k)

Cumulative distribution function

F(x) = \alpha_1 F_1(x; \lambda_1) + \dots + \alpha_k F_k(x; \lambda_k)

where \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

Details

Probability density function

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

Cumulative distribution function

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

Quantile function

F^{-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

Details

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

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

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

Details

Probability density function

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

Cumulative distribution function

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

Quantile function

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

Details

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

Probability density function

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(-\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

Details

If X and Y follow Poisson distributions with means \mu_1 and \mu_2, than X-Y follows Skellam distribution parametrized by \mu_1 and \mu_2.

Probability mass function

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

Details

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

Probability density function

f(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) = \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

Details

Probability density function

f(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) = \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) = \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

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

Details

Probability density function

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

Cumulative distribution function

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) = \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

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

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) = \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

Details

Probability density function

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) = \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

Details

Probability density function

f(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

Details

Probability density function

f(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

Details

Probability density function

f(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")