dPL {RelDists}

The Power Lindley distribution

Description

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.

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.

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.

Author(s)

Amylkar Urrea Montoya, [e-mail]

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.

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