Cumulative Distribution Function of the Hurdle Negative Binomial Distribution

Computes the cumulative distribution function (CDF) or its logarithm for the hurdle negative binomial (HNB) distribution. The hurdle model combines a point mass at zero with a truncated negative binomial distribution for positive counts.

Usage

phurdle.nb(y, mu, size, pi, lower.tail = FALSE, log.p = FALSE)

Arguments

y: Numeric vector of observed count values.
mu: Numeric vector of mean parameters of the negative binomial distribution.
size: Numeric vector of shape (dispersion) parameters of the negative binomial distribution.
pi: Numeric vector of hurdle probabilities (probability of structural zeros).
lower.tail: Logical; if TRUE (default), probabilities are $P(Y \le y)$; otherwise, they are $P(Y > y)$.
log.p: Logical; if TRUE, probabilities are returned on the log scale.

Value

A numeric vector of cumulative probabilities (or log-probabilities if log.p = TRUE).

Details

The hurdle negative binomial model assumes: $$ P(Y = 0) = \pi, \quad P(Y = y \mid Y > 0) = (1 - \pi) \frac{F_{NB}(y)-F_{NB}(0)}{1 - F_{NB}(0)}, \quad y > 0 $$ where $F_{NB}(y)$ is CDF of the standard negative binomial distribution.

The function computes the upper or lower tail probabilities for both zeros and positive counts using the logarithmic form for numerical stability. Internal helper functions (log_diff_exp, log_sum_exp) are used to handle differences and sums of log-scale probabilities safely.

Examples

# Example: Hurdle Negative Binomial CDF
y <- 0:5
mu <- 2
size <- 1.5
pi <- 0.3
phurdle.nb(y, mu, size, pi)
#> [1] 0.70000000 0.46601122 0.29887638 0.18745315 0.11582394 0.07079986

# Upper tail probabilities on log scale
phurdle.nb(y, mu, size, pi, lower.tail = FALSE, log.p = TRUE)
#> [1] -0.3566749 -0.7635456 -1.2077252 -1.6742263 -2.1556840 -2.6478983