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