QLindley: Quasi 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 quasi Lindley distribution with parameters theta and alpha.

Usage

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

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

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

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

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

Arguments

x, q	vector of positive quantiles.
theta	positive parameter.
alpha	greater than -1.
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 ≤ft( α +θ x\right) {{e}^{-θ x}}}{1+α }}

Cumulative distribution function

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

Quantile function

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

Hazard rate function

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

where W_{-1} denotes the negative branch of the Lambert W function.

Particular cases: α = θ the one-parameter Lindley distribution and for α=0 the gamma distribution with shape = 2 and scale = θ.

Value

dqlindley gives the density, pqlindley gives the distribution function, qqlindley gives the quantile function, rqlindley generates random deviates and hqlindley 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]qlindley are calculated directly from the definitions. rqlindley uses either a two-component mixture of gamma distributions or the quantile function.

References

Shanker, R. and Mishra, A. (2013). A quasi Lindley distribution. African Journal of Mathematics and Computer Science Research, 6, (4), 64-71.

See Also

lambertWm1.

Examples

set.seed(1)
x <- rqlindley(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, dqlindley(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)
pqlindley(q, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
pqlindley(q, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
qqlindley(p, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
qqlindley(p, theta = 1.5, alpha = 1.5, lower.tail = FALSE)

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

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