# LindleyE: Lindley Exponential 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 Lindley exponential distribution with parameters theta and alpha.

## Usage

 1 2 3 4 5 6 7 8 9 dlindleye(x, theta, alpha, log = FALSE) plindleye(q, theta, alpha, lower.tail = TRUE, log.p = FALSE) qlindleye(p, theta, alpha, lower.tail = TRUE, log.p = FALSE) rlindleye(n, theta, alpha) hlindleye(x, theta, alpha, log = FALSE) 

## Arguments

 x, q vector of positive quantiles. theta, alpha 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.

## Details

Probability density function

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

Cumulative distribution function

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

Quantile function

\code{see Bhati et al., 2015}

Hazard rate function

\code{see Bhati et al., 2015}

## Value

dlindleye gives the density, plindleye gives the distribution function, qlindleye gives the quantile function, rlindleye generates random deviates and hlindleye 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]lindleye are calculated directly from the definitions. rlindleye uses the quantile function.

## References

Bhati, D., Malik, M. A., Vaman, H. J., (2015). Lindley-Exponential distribution: properties and applications. METRON, 73, (3), 335–357.

lambertWm1.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 set.seed(1) x <- rlindleye(n = 1000, theta = 5.0, alpha = 0.2) R <- range(x) S <- seq(from = R[1], to = R[2], by = 0.1) plot(S, dlindleye(S, theta = 5.0, alpha = 0.2), 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) plindleye(q, theta = 5.0, alpha = 0.2, lower.tail = TRUE) plindleye(q, theta = 5.0, alpha = 0.2, lower.tail = FALSE) qlindleye(p, theta = 5.0, alpha = 0.2, lower.tail = TRUE) qlindleye(p, theta = 5.0, alpha = 0.2, lower.tail = FALSE) ## waiting times data (from Ghitany et al., 2008) data(waitingtimes) library(fitdistrplus) fit <- fitdist(waitingtimes, 'lindleye', start = list(theta = 2.6, alpha = 0.15), lower = c(0.01, 0.01)) plot(fit)