Cumulative Distribution Function (CDF) of Zero-Truncated Negative Binomial Distribution

Computes the cumulative distribution function (CDF) for a zero-truncated negative binomial (TNB) distribution. This function adjusts the standard negative binomial CDF to account for truncation at zero (i.e., only supports \(y > 0\)).

Usage

cdf.tnb(y, mu, size, lower.tail = FALSE, log.p = FALSE)

Arguments

y

Numeric vector of quantiles (count values) for which to compute the CDF.

mu

Mean parameter (\(\mu\)) of the negative binomial distribution.

size

Dispersion (shape) parameter (\(r\)) of the negative binomial distribution.

lower.tail

Logical; if TRUE (default), probabilities are \(P(Y \le y)\). If FALSE, probabilities are \(P(Y > y)\).

log.p

Logical; if TRUE, probabilities \(p\) are given as \(\log(p)\).

Value

A numeric vector of the same length as the input, containing:

• CDF values (\(P(Y \le y \mid Y > 0)\)) if lower.tail = TRUE.

• Upper-tail probabilities (\(P(Y > y \mid Y > 0)\)) if lower.tail = FALSE.

• Log-probabilities if log.p = TRUE.

Details

The function computes probabilities for the zero-truncated version of the negative binomial distribution: $$P(Y \le y \mid Y > 0) = \frac{P(Y \le y) - P(Y = 0)}{1 - P(Y = 0)}.$$

Internally, this is implemented using the log-scale for numerical stability. When lower.tail = FALSE, it computes the upper-tail probabilities \(P(Y > y \mid Y > 0)\) instead.

See also

pnbinom for the standard negative binomial CDF.

Examples

# Example: Compute the CDF for y = 1:5
mu <- 2
size <- 1
cdf.tnb(1:5, mu, size)
#> [1] 0.6666667 0.4444444 0.2962963 0.1975309 0.1316872

# Compute the upper-tail probabilities
cdf.tnb(1:5, mu, size, lower.tail = FALSE)
#> [1] 0.6666667 0.4444444 0.2962963 0.1975309 0.1316872

# Log probabilities
cdf.tnb(1:5, mu, size, log.p = TRUE)
#> [1] -0.4054651 -0.8109302 -1.2163953 -1.6218604 -2.0273255