EXPLindley: Exponentiated 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 exponentiated Lindley distribution with parameters theta and alpha.

Usage

 1 2 3 4 5 6 7 8 9 dexplindley(x, theta, alpha, log = FALSE) pexplindley(q, theta, alpha, lower.tail = TRUE, log.p = FALSE) qexplindley(p, theta, alpha, lower.tail = TRUE, log.p = FALSE) rexplindley(n, theta, alpha) hexplindley(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+θ )} (1+x)e^{-θ x}≤ft[ 1-≤ft( 1+\frac{θ x}{1+θ }\right) e^{-θ x}\right] ^{α -1}

Cumulative distribution function

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

Quantile function

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

Hazard rate function

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

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

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

Value

dexplindley gives the density, pexplindley gives the distribution function, qexplindley gives the quantile function, rexplindley generates random deviates and hexplindley gives the hazard rate function.

Invalid arguments will return an error message.

Note

Nadarajah et al. (2011) named the exponentiated Lindley distribution as generalized Lindley distribution.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Larissa B. Fernandes lbf.estatistica@gmail.com

Source

[d-h-p-q-r]explindley are calculated directly from the definitions. rexplindley uses the quantile function.

References

Nadarajah, S., Bakouch, H. S., Tahmasbi, R., (2011). A generalized Lindley distribution. Sankhya B, 73, (2), 331-359.

lambertWm1.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 set.seed(1) x <- rexplindley(n = 1000, theta = 1.5, alpha = 1.5) R <- range(x) S <- seq(from = R[1], to = R[2], by = 0.1) plot(S, dexplindley(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) pexplindley(q, theta = 1.5, alpha = 1.5, lower.tail = TRUE) pexplindley(q, theta = 1.5, alpha = 1.5, lower.tail = FALSE) qexplindley(p, theta = 1.5, alpha = 1.5, lower.tail = TRUE) qexplindley(p, theta = 1.5, alpha = 1.5, lower.tail = FALSE) ## Relief times data (from Nadarajah et al., 2011) data(relieftimes) library(fitdistrplus) fit <- fitdist(relieftimes, 'explindley', start = list(theta = 1.5, alpha = 1.5)) plot(fit)