# 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

 1 2 3 4 5 6 7 8 9 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 [email protected]

Larissa B. Fernandes [email protected]

## 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.

lambertWm1.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 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)