SLindley: Two-Parameter Lindley Distribution

In LindleyR: The Lindley Distribution and Its Modifications

Description

Density function, distribution function, quantile function, random number generation and hazard rate function for the two-parameter Lindley distribution with parameters theta and alpha.

Usage

dslindley(x, theta, alpha, log = FALSE)

pslindley(q, theta, alpha, lower.tail = TRUE, log.p = FALSE)

qslindley(p, theta, alpha, lower.tail = TRUE, log.p = FALSE)

rslindley(n, theta, alpha, mixture = TRUE)

hslindley(x, theta, alpha, log = FALSE)

Arguments

x, q	vector of positive quantiles.
theta	positive parameter.
alpha	greater than -theta.
log, log.p	logical; If TRUE, probabilities p are given as log(p).
lower.tail	logical; If TRUE, (default), P(X ≤q x) are returned, otherwise P(X > x).
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.
mixture	logical; If TRUE, (default), random deviates are generated from a two-component mixture of gamma distributions, otherwise from the quantile function.

Details

Probability density function

f(x\mid θ,α )=\frac{{θ }^{2}}{θ +α }≤ft(1+α x\right) e^{-θ x}

Cumulative distribution function

F(x\mid θ,α )=1-\frac{≤ft( θ + α +α θ x\right) }{θ +α }e^{-θ x}

Quantile function

Q(p\mid θ,α )=-\frac{1}{θ }-\frac{1}{α }-\frac{1}{θ }W_{-1}≤ft( \frac{1}{α }(p-1)≤ft( θ +α \right)e^{-{\frac{α +θ }{α }}}\right)

Hazard rate function

h(x\mid θ )=\frac{θ ^{2}}{≤ft( θ + α +αθ x\right) }(1+α x)

where θ > 0, α > -θ and W_{-1} denotes the negative branch of the Lambert W function.

Particular case: α = 1 the one-parameter Lindley distribution.

Value

dslindley gives the density, pslindley gives the distribution function, qslindley gives the quantile function, rslindley generates random deviates and hslindley gives the hazard rate function.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli [e-mail]

Larissa B. Fernandes [e-mail]

Source

[d-h-p-q-r]slindley are calculated directly from the definitions. rslindley uses either a two-component mixture of the gamma distributions or the quantile function.

References

Shanker, R., Sharma, S. and Shanker, R. (2013). A two-parameter Lindley distribution for modeling waiting and survival times data. Applied Mathematics, 4, (2), 363-368.

See Also

lambertWm1.

Examples

set.seed(1)
x <- rslindley(n = 1000, theta = 1.5, alpha = 1.5, mixture = TRUE)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.1)
plot(S, dslindley(S, theta = 1.5, alpha = 1.5), xlab = 'x', ylab = 'pdf')
hist(x, prob = TRUE, main = '', add = TRUE)

p <- seq(from = 0.1, to = 0.9, by = 0.1)
q <- quantile(x, prob = p)
pslindley(q, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
pslindley(q, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
qslindley(p, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
qslindley(p, theta = 1.5, alpha = 1.5, lower.tail = FALSE)

library(fitdistrplus)
fit <- fitdist(x, 'slindley', start = list(theta = 1.5, alpha = 1.5))
plot(fit)

LindleyR documentation built on May 1, 2019, 8:05 p.m.