Lindley: One-Parameter Lindley Distribution

In flexCountReg: Estimation of a Variety of Count Regression Models

One-Parameter Lindley Distribution

Description

Distribution function for the one-parameter Lindley distribution with parameter theta.

Usage

dlindley(x, theta = 1, log = FALSE)

plindley(q, theta = 1, lower.tail = TRUE, log.p = FALSE)

qlindley(p, theta = 1, lower.tail = TRUE, log.p = FALSE)

rlindley(n, theta = 1)

Arguments

x	a single value or vector of positive values.
theta	distribution parameter value. Default is 1.
log, log.p	logical; If TRUE, probabilities p are given as log(p). If FALSE, probabilities p are given directly. Default is FALSE.
q	a single value or vector of quantiles.
lower.tail	logical; If TRUE, (default), P(X \leq x) are returned, otherwise P(X > x) is returned. Default is TRUE.
p	a single value or vector of probabilities.
n	number of random values to generate.

Details

Probability density function (PDF)

f(x\mid \theta )=\frac{\theta ^{2}}{(1+\theta )}(1+x)e^{-\theta x}

Cumulative distribution function (CDF)

F(x\mid \theta ) = 1 - \left(1+ \frac{\theta x}{1+\theta }\right)e^{-\theta x}

Quantile function (Inverse CDF)

Q(p\mid\theta) = -1 - \frac{1}{\theta} - \frac{1}{\theta} W_{-1}\!\left((1+\theta)(p-1)e^{-(1+\theta)}\right)

where W_{-1}() is the negative branch of the Lambert W function.

The moment generating function (MGF) is:

M_X(t)=\frac{\theta^2(\theta-t+1)}{(\theta+1)(\theta-t)^2}

The distribution mean and variance are:

\mu=\frac{\theta+2}{\theta(1+\theta)}

\sigma^2=\frac{\mu}{\theta+2}\left(\frac{6}{\theta}-4\right)-\mu^2

Value

dlindley gives the density, plindley gives the distribution function, qlindley gives the quantile function, and rlindley generates random deviates.

The length of the result is determined by n for rlindley, and is the maximum of the lengths of the numerical arguments for the other functions.

Examples

x <- seq(0, 5, by = 0.1)
p <- seq(0.1, 0.9, by = 0.1)
q <- c(0.2, 3, 0.2)
dlindley(x, theta = 1.5)
dlindley(x, theta=0.5, log=TRUE)
plindley(q, theta = 1.5)
plindley(q, theta = 0.5, lower.tail = FALSE)
qlindley(p, theta = 1.5)
qlindley(p, theta = 0.5)

set.seed(123154)
rlindley(5, theta = 1.5)
rlindley(5, theta = 0.5)

flexCountReg documentation built on Jan. 20, 2026, 1:06 a.m.