RelDists (version 1.0.2)

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

Description

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)

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.

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.

Author

Sebastián Ándres Rios Romero,

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.

See Also

IPXLIN.

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)