dPoisGam {fishMod}

Density, derivatives, distribution function, and random generation for the Poisson-Gamma distribution.

Description

Density, derivatives, distribution function, and random generation for the Poisson-Gamma distribution. The distribution is parameterised by the Poisson mean, the Gamma mean and the Gamma dispersion parameter. Derivatives are with respect to these three parameters.

Usage

dPoisGam( y, lambda, mu.Z, alpha, LOG=TRUE)
dPoisGamDerivs( y, lambda, mu.Z, alpha, do.checks=TRUE)
pPoisGam( q, lambda, mu.Z, alpha)
rPoisGam( n, lambda, mu.Z, alpha)

Arguments

y, q

vector of quantiles

n

number of random draws

lambda

scalar or vector (length matches y or equal to n) of Poisson means

mu.Z

scalar or vector (length matches y or equal to n) of Gamma means

alpha

scalar or vector (length matches y or equal to n) of Gamma dispersions

LOG

indication of return scale. If TRUE (default) then the density is returned on the log scale.

do.checks

boolean indicating if checks on arguments should be performed. If TRUE (default) then checks will be performed.

Details

The observed random variables y_i are assumed to arise from the process: y_i=sum(z_{i1}+z_{i2}+...+z_{in_i}) where n_i is a Poisson variable with mean lambda and z_{ij} is a Gamma variable with mean mu.Z and variance mu.Z^2 / alpha.

The density calculation is based on the series summation method described in Dunn and Smyth (2005). However, the parameterisation used here is different and follows that described in Section 2 of Smyth (1996). The details of density calculation are subsequently different from Dunn and Smyth (2005).

The derivatives are calculated in a similar manner to the density. The derivatives are for the log-density.

Value

dPoisGam returns a numeric vector containing the (log-)densities.

dPoisGamDeriv returns a matrix with three columns, one for each of the distributional parameters.

pPoisGam returns a numeric vector containing the values of the distribution function.

rPoisGam returns a numeric vector containing the random variables.

Author(s)

Scott D. Foster

References

Smyth, G. K. (1996) Regression analysis of quantity data with exact zeros. Proceedings of the Second Australia–Japan Workshop on Stochastic Models in Engineering, Technology and Management. Technology Management Centre, University of Queensland, pp. 572-580.

Dunn P. K. and Smyth G. K. (2005) Series evaluation of Tweedie exponential dispersion model densities. Statistics and Computing 15: 267-280.

Foster, S.D. and Bravington, M.V. (2013) A Poisson-Gamma Model for Analysis of Ecological Non-Negative Continuous Data. Journal of Environmental and Ecological Statistics 20: 533-552

Examples

my.seq <- seq( from=0, to=20, length=200)
par( mfrow=c( 1,2))
plot( my.seq, dPoisGam( y=my.seq, lambda=8, mu.Z=1.2, alpha=0.6, LOG=FALSE), type='l',
 xlab="Variable", ylab="Density")
plot( my.seq, pPoisGam( q=my.seq, lambda=8, mu.Z=1.2, alpha=0.6), type='l',
 xlab="Variable", ylab="Distribution")