flexCountReg (version 0.1.1)

dlindley: 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)

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.

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: $$\mu=\frac{\theta+2}{\theta(1+\theta)}$$ $$\sigma^2=\frac{\mu}{\theta+2}\left(\frac{6}{\theta}-4\right)-\mu^2$$

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)