Cumulative Distribution Function of the Hurdle Poisson Distribution

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

Usage

phurdle.pois(y, lambda, pi, lower.tail = FALSE, log.p = FALSE)

Arguments

y: Numeric vector of observed count values.
lambda: Numeric vector of mean parameters of the poisson 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 poisson model assumes: $$ P(Y = 0) = \pi, \quad P(Y = y \mid Y > 0) = (1 - \pi) \frac{F_{Pois}(y)-F_{Pois}(0)}{1 - F_{Pois}(0)}, \quad y > 0 $$ where $F_{Pois}(y)$ is CDF of the standard possion 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 Poisson CDF
y <- 0:5
lambda <- 2
pi <- 0.3
phurdle.pois(y, lambda, pi)
#> [1] 0.70000000 0.48087530 0.26175060 0.11566747 0.04262590 0.01340927

# Upper tail probabilities on log scale
phurdle.pois(y, lambda, pi, lower.tail = FALSE, log.p = TRUE)
#> [1] -0.3566749 -0.7321473 -1.3403631 -2.1570359 -3.1552932 -4.3118087