The Inverse Power XLindley distribution - Hassan et al. (2025)

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

Usage

dIPXLIN(x, mu = 1, sigma = 1, log = FALSE)

pIPXLIN(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qIPXLIN(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rIPXLIN(n, mu = 1, sigma = 1)

hIPXLIN(x, mu = 1, sigma = 1)

Arguments

x, q: vector of quantiles.
mu: parameter representing \(\eta\) (mu > 0).
sigma: parameter representing \(\sigma\) (sigma > 0).
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

dIPXLIN gives the density, pIPXLIN gives the distribution function, qIPXLIN gives the quantile function, rIPXLIN generates random deviates and hIPXLIN gives the hazard function.

Details

The Inverse Power XLindley with parameters mu and sigma has density given by

\(f(x|\mu,\sigma) = \frac{\sigma \mu^2}{(1+\mu)^2} x^{-2\sigma-1} \left(1 + (2+\mu) x^\sigma\right) e^{-\mu x^{-\sigma}}\)

for \(x>0\), \(\mu>0\) and \(\sigma>0\). In this parameterization, \(\mu\) is the scale parameter and \(\sigma\) is the shape parameter.

References

Hassan, A. S., Alsadat, N., Chesneau, C., Elgarhy, M., Kayid, M., Nasiru, S., & Gemeay, A. M. (2025). Inverse power XLindley distribution with statistical inference and applications to engineering data. Scientific Reports, 15, 4385.

Author

Sebastián Ándres Rios Romero, srios.romero@udea.edu.co

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dIPXLIN(x, mu=0.5, sigma=1.5), 
      from=0.001, to=2.5,
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dIPXLIN(x, mu=1.5, sigma=3.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=1.5", 
                            "mu=1.5, sigma=3.5"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.6)



# Example 2
# Checking if the cumulative curves converge to 1
curve(pIPXLIN(x, mu=0.5, sigma=1.5), 
      from=0.00001, to=4,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="F(x)")
curve(pIPXLIN(x, mu=1.5, sigma=4.0),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=0.5, sigma=1.5", 
                               "mu=1.5, sigma=4.0"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.5)



# Example 3 
p <- seq(from=0, to=0.99, length.out=100)
plot(x=qIPXLIN(p, mu=0.5, sigma=1.5), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pIPXLIN(x, mu=0.5, sigma=1.5), 
      from=0, add=TRUE, col="tomato", lwd=2.5)



# Example 4
# The random function
x <- rIPXLIN(n=1000, mu=0.5, sigma=3.5)
hist(x, freq=FALSE, breaks=50, xlim=c(0,4))
curve(dIPXLIN(x, mu=0.5, sigma=3.5),  
      add=TRUE, col="tomato", lwd=2)



# Example 5
# The Hazard function
curve(hIPXLIN(x, mu=0.5, sigma=1.5), from=0.001, to=4,
      col="tomato", ylab="Hazard function", las=1)