compound {gentransmuted}R Documentation

The Exponentiated and Exponentiated2 distributions

Description

Density, distribution function, quantile function and random generation for the compound distributions.

Usage

dcompound(x, dist = "exp", comp1 = as.null(), comp2 = as.null(), gamma = 1, beta = 1, 
	theta1 = 1, theta2 = 1, log = FALSE)
pcompound(q, dist = "exp", comp1 = as.null(), comp2 = as.null(), gamma = 1, beta = 1, 
	theta1 = 1, theta2 = 1, lower.tail = TRUE, log.p = FALSE)
qcompound(p, dist = "exp", comp1 = as.null(), comp2 = as.null(), gamma = 1, beta = 1, 
	theta1 = 1, theta2 = 1, lower.tail = TRUE, log.p = FALSE)
rcompound(n, dist = "exp", comp1 = as.null(), comp2 = as.null(), gamma = 1, beta = 1, 
	theta1 = 1, theta2 = 1)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

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

dist

baseline distribution. Avaliable options: exp (exponential), gamma, lnorm (log-normal), paretoII, bisa (Birnbaum-Saunders), lomax, beta, kumar (Kumaraswamy), norm (normal), logis (logistic), cauchy, gumbel. See details for parameterizations of these distributions.

comp1, comp2

compounding distributions. Avaliable options: EXP (Exponentiated), EXP2 (Exponentiated of the second kind), MO (Marshall-Olkin), MO2 (Marshall-Olkin of the second kind), SB (Shaw and Buckley).

gamma, beta

parameters for the baseline distribution.

theta1, theta2

shape parameter for the comp1 and comp2 distributions, respectively.

log, log.p

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

lower.tail

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

Details

The compound distribution has cumulative distribution function

F(x;\gamma,\beta,\theta_1,\theta_2)=G_2(G_1(F(x;\gamma,\beta),\theta_1),\theta_2),

where F is related to dist, G_1 is related to comp1 and G_2 is related to comp2. The support for x depends on the baseline distribution. For exp, gamma, lnorm, paretoII, bisa and lomax, the support is (0,\infty); for beta and kumar is (0,1); for norm, logis, cauchy and gumbel is (-\infty,\infty). The parameter space for \gamma and \beta also depend on the baseline distribution. For exp, \gamma>0; for gamma, paretoII, bisa, lomax, beta and kumar \gamma,\beta >0; for lnorm, norm, logis, cauchy and gumbel \gamma \in \mathbb{R}, \beta>0. The parameter space for \theta_1 and \theta_2 depend on comp1 and comp2. For EXP, EXP2, MO and MO2 options the corresponding parameter space is (0,\infty), whereas for SB option is (-1,1). The probability density function for each of the baseline distribution is given below. exp

f(x)=\gamma e^{-\gamma x}

gamma

f(x)=\frac{\beta^\gamma}{\Gamma(\gamma)}x^{\gamma-1} e^{-\beta x}

lnorm

f(x)=\frac{1}{x\sqrt{2\pi \beta^2}}\exp\left(-\frac{(\log(x)-\gamma)^2}{\beta^2}\right)

paretoII

f(x)=\frac{\gamma}{\beta}\left(1+\frac{x}{\beta}\right)^{-(\gamma+1)}

bisa

f(x)=\frac{1}{2\sqrt{2\pi}\gamma\beta}\left[\left(\frac{\beta}{x}\right)^{1/2}+\left(\frac{\beta}{x}\right)^{3/2}\right]\exp\left[-\frac{1}{2\gamma^2}\left(\frac{x}{\beta}+\frac{\beta}{x}-2\right)\right]

beta

f(x)=\frac{1}{B(\gamma,\beta)}x^{\gamma-1}(1-x)^{\beta-1}

kumar

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

For norm, logis, cauchy and gumbel, the probability density function is given by

f(x)=\frac{1}{\beta} g\left(\frac{x-\gamma}{\beta}\right)

where g is given by norm

g(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}

logis

g(x)=\frac{e^{x}}{(1+e^x)^2}

cauchy

g(x)=\frac{1}{\pi (1+x^2)}

gumbel

g(x)=\exp\left(-(x+e^{-x})\right)

Value

dcompound gives the density, pcompound gives the distribution function, qcompound gives the quantile function, and rcompound generates random deviates. The length of the result is determined by n for rcompound, and is the maximum of the lengths of the numerical arguments for the other functions. The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Author(s)

Yolanda M. Gomez, Diego I. Gallardo, Hector W. Gomez and Barry Arnold

Examples

set.seed(2100)
y=rcompound(100, 1.2, 1.4, 1, 0.8, dist="exp", comp1="EXP", comp2="MO")

[Package gentransmuted version 1.0 Index]