Extended Power Lindley Distribution

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Description

Density function, distribution function, quantile function, random number generation and hazard rate function for the extended power Lindley distribution with parameters theta, alpha and beta.

Usage

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dextplindley(x, theta, alpha, beta, log = FALSE)

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

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

rextplindley(n, theta, alpha, beta, mixture = TRUE)

hextplindley(x, theta, alpha, beta, log = FALSE)

Arguments

x, q

vector of positive quantiles.

theta, alpha, beta

positive parameters.

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,β )={\frac{α θ ^{2}}{θ +β }}(1+β x^{α })\ x^{α -1}\ e^{-θ x^{α }}

Cumulative distribution function

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

Quantile function

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

Hazard rate function

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

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

Particular cases: β = 1 the power Lindley distribution, α = 1 the two-parameter Lindley distribution and (α = 1, β = 1) the one-parameter Lindley distribution.

Value

dextplindley gives the density, pextplindley gives the distribution function, qextplindley gives the quantile function, rextplindley generates random deviates and hextplindley gives the hazard rate function.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Larissa B. Fernandes lbf.estatistica@gmail.com

Source

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

References

Alkarni, S. H., (2015). Extended power Lindley distribution: A new statistical model for non-monotone survival data. European Journal of Statistics and Probability, 3, (3), 19-34.

See Also

lambertWm1.

Examples

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set.seed(1)
x <- rextplindley(n = 1000, theta = 1.5, alpha = 1.5, beta = 1.5, mixture = TRUE)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.1)
plot(S, dextplindley(S, theta = 1.5, alpha = 1.5, beta = 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)
pextplindley(q, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = TRUE)
pextplindley(q, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = FALSE)
qextplindley(p, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = TRUE)
qextplindley(p, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = FALSE)

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