countreg

Extension of the Zero-Truncated Poisson Distribution

Description

Score function, hessian, mean, and variance for the zero-truncated Poisson distribution with parameter lambda (= mean of the untruncated distribution) or mean (= of the truncated distribution).

Usage

sztpois(x, lambda, mean, parameter = "lambda", drop = TRUE)
hztpois(x, lambda, mean, parameter = "lambda", drop = TRUE)
mean_ztpois(lambda, mean, drop = TRUE)
var_ztpois(lambda, mean, drop = TRUE)

Arguments

`x`	vector of (positive integer) quantiles.
`lambda`	vector of (non-negative) means of the untruncated Poisson distribution. Only one of `lambda` or `mean` should be specified.
`mean`	vector of means (greater than 1) of the zero-truncated Poisson distribution. Only one of `lambda` or `mean` should be specified.
`parameter`	character. Should the derivative with respect to `“lambda”` or `“mean”` be computed?
`drop`	logical. Should the result be a matrix (`drop = FALSE`) or should the dimension be dropped (`drop = TRUE`, the default)?

Details

The untruncated Poisson distribution has density

\(f(x) = \frac{\lambda^x e^{-\lambda}}{x!}\)

for \(x = 0, 1, 2, \ldots\). The zero-truncated density is then simply obtained as

\(g(x) = \frac{f(x)}{1 - f(0)}\)

for \(x = 1, 2, \ldots\).

The zero-truncated distribution has expectation \(E(X) = \mu = \lambda / (1 - \exp(-\lambda))\) and variance \(Var(X) = \mu \cdot (\lambda + 1 - \mu)\), where \(\lambda\) is the expectation of the untruncated Poisson distribution.

Despite the simple form of the transformation \(\mu(\lambda)\) the inverse \(\lambda(\mu)\) has no closed-form solution and is computed numerically if needed.

Value

sztpois gives the score function (= derivative of the log-density with respect to lambda or mean). hztpois gives the hessian (= 2nd derivative of the log-density with respect to lambda or mean). mean_ztpois and var_ztpois give the mean and the variance, respectively.