The Power Lindley distribution

Density, distribution function, quantile function, random generation and hazard function for the Power Lindley distribution with parameters mu and sigma.

Usage

dPL(x, mu, sigma, log = FALSE)

pPL(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)

qPL(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)

rPL(n, mu, sigma)

hPL(x, mu, sigma)

Arguments

x, q: vector of quantiles.
mu: parameter.
sigma: parameter.
log, log.p: logical; if TRUE, probabilities p are given as log(p).
lower.tail: logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
p: vector of probabilities.
n: number of observations.

Value

dPL gives the density, pPL gives the distribution function, qPL gives the quantile function, rPL generates random deviates and hPL gives the hazard function.

Details

The Power Lindley Distribution with parameters mu and sigma has density given by

\(f(x) = \frac{\mu \sigma^2}{\sigma + 1} (1 + x^\mu) x ^ {\mu - 1} \exp({-\sigma x ^\mu}),\)

for x > 0.

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010 .

Ghitanya ME, Al-Mutairi DK, Balakrishnanb N, Al-Enezi LJ (2013). “Power Lindley distribution and associated inference.” Computational Statistics and Data Analysis, 64, 20–33. doi:10.1016/j.csda.2013.02.026 .

Author

Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function
curve(dPL(x, mu=1.5, sigma=0.2), from=0.1, to=10,
      col="red", las=1, ylab="f(x)")


## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pPL(x, mu=1.5, sigma=0.2),
      from=0.1, to=10, col="red", las=1, ylab="F(x)")
curve(pPL(x, mu=1.5, sigma=0.2, lower.tail=FALSE),
      from=0.1, to=10, col="red", las=1, ylab="R(x)")


## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qPL(p, mu=1.5, sigma=0.2), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pPL(x, mu=1.5, sigma=0.2), from=0.1, add=TRUE, col="red")

## The random function
hist(rPL(n=1000, mu=1.5, sigma=0.2), freq=FALSE,
     xlab="x", las=1, main="")
curve(dPL(x, mu=1.5, sigma=0.2), from=0.1, to=15, add=TRUE, col="red")


## The Hazard function
par(mfrow=c(1,1))
curve(hPL(x, mu=1.5, sigma=0.2), from=0.1, to=15,
      col="red", ylab="Hazard function", las=1)


par(old_par) # restore previous graphical parameters