countreg

Extension of the Hurdle Negative Binomial Distribution

Description

Score function, hessian, mean, and, variance for the (zero-)hurdle negative binomial distribution with parameters mu (= mean of the underlying negative binomial distribution), dispersion parameter theta (or equivalently size), and hurdle crossing probability pi (i.e., 1 - pi is the probability for observed zeros).

Usage

shnbinom(x, mu, theta, size, pi, parameter = c("mu", "theta", "pi"), drop = TRUE)
hhnbinom(x, mu, theta, size, pi, parameter = c("mu", "theta", "pi"), drop = TRUE)
mean_hnbinom(mu, theta, size, pi, drop = TRUE)
var_hnbinom(mu, theta, size, pi, drop = TRUE)

Arguments

`x`	vector of (positive integer) quantiles.
`mu`	vector of non-negative means of the underlying negative binomial distribution.
`theta`, `size`	vector of strictly positive dispersion parameters (shape parameter of the gamma mixing distribution). Only one of `theta` or `size` must be specified.
`pi`	vector of hurdle crossing probabilities (i.e., `1 - pi` is the probability for observed zeros).
`parameter`	character. Should the derivative with respect to `“mu”` and/or `“theta”` and/or `“pi”` be computed?
`drop`	logical. Should the result be a matrix (`drop = FALSE`) or should the dimension be dropped (`drop = TRUE`, the default)?

Details

The underlying negative binomial distribution has density

f(x) =

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

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

for \(x = 1, 2, \ldots\) and \(g(0) = 1 - \pi\), respectively.

Value

shnbinom gives the score function (= derivative of the log-density with respect to mu and/or theta and/or pi). hhnbinom gives the hessian (= 2nd derivative of the log-density with respect to mu and/or theta and/or pi). mean_hnbinom and var_hnbinom give the mean and the variance, respectively.