# MOLindley: Marshall-Olkin Extended 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 Marshall-Olkin extended Lindley distribution with parameters theta and alpha.

## Usage

 1 2 3 4 5 6 7 8 9 dmolindley(x, theta, alpha, log = FALSE) pmolindley(q, theta, alpha, lower.tail = TRUE, log.p = FALSE) qmolindley(p, theta, alpha, lower.tail = TRUE, log.p = FALSE) rmolindley(n, theta, alpha) hmolindley(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{α θ^{2}(1+x)e^{-θ x}}{(1+θ )≤ft[ 1-\overline{α }≤ft( 1+\frac{θ x}{1+θ }\right) e^{-θ x}\right] ^{2}}

Cumulative distribution function

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

Quantile function

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

Hazard rate function

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

where \overline{α}=(1 - α) and W_{-1} denotes the negative branch of the Lambert W function.

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

## Value

dmolindley gives the density, pmolindley gives the distribution function, qmolindley gives the quantile function, rmolindley generates random deviates and hmolindley 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]molindley are calculated directly from the definitions. rmolindley uses the quantile function.

## References

do Espirito Santo, A. P. J., Mazucheli, J., (2015). Comparison of estimation methods for the Marshall-Olkin extended Lindley distribution. Journal of Statistical Computation and Simulation, 85, (17), 3437-3450.

Ghitany, M. E., Al-Mutairi, D. K., Al-Awadhi, F. A. and Al-Burais, M. M., (2012). Marshall-Olkin extended Lindley distribution and its application. International Journal of Applied Mathematics, 25, (5), 709-721.

Marshall, A. W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84, (3), 641.652.

lambertWm1, Lindley.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 set.seed(1) x <- rmolindley(n = 1000, theta = 5, alpha = 5) R <- range(x) S <- seq(from = R[1], to = R[2], by = 0.1) plot(S, dmolindley(S, theta = 5, alpha = 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) pmolindley(q, theta = 5, alpha = 5, lower.tail = TRUE) pmolindley(q, theta = 5, alpha = 5, lower.tail = FALSE) qmolindley(p, theta = 5, alpha = 5, lower.tail = TRUE) qmolindley(p, theta = 5, alpha = 5, lower.tail = FALSE) ## bladder cancer data (from Warahena-Liyanage and Pararai, 2014) data(bladdercancer) library(fitdistrplus) fit <- fitdist(bladdercancer, 'molindley', start = list(theta = 0.1, alpha = 1.0)) plot(fit)