RelDists (version 1.0.1)

dEXL: The exponentiated XLindley distribution

Description

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

Usage

dEXL(x, mu, sigma, log = FALSE)
pEXL(q, mu, sigma, log.p = FALSE, lower.tail = TRUE)
qEXL(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rEXL(n, mu, sigma)
hEXL(x, mu, sigma, log = FALSE)

Value

dEXL gives the density, pEXL gives the distribution function, qEXL gives the quantile function, rEXL

generates random deviates and hEXL gives the hazard function.

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.

Author

Manuel Gutierrez Tangarife,

Details

The exponentiated XLindley with parameters mu and sigma has density given by

\( f(x) = \frac{\sigma\mu^2(2+\mu + x)\exp(-\mu x)}{(1+\mu)^2}\left[1- \left(1+\frac{\mu x}{(1 + \mu)^2}\right) \exp(-\mu x)\right] ^ {\sigma-1} \)

for \(x \geq 0\), \(\mu \geq 0\) and \(\sigma \geq 0\).

Note: In this implementation we changed the original parameters \(\delta\) for \(\mu\) and \(\alpha\) for \(\sigma\), we did it to implement this distribution within gamlss framework.

References

Alomair, A. M., Ahmed, M., Tariq, S., Ahsan-ul-Haq, M., & Talib, J. (2024). An exponentiated XLindley distribution with properties, inference and applications. Heliyon, 10(3).

See Also

EXL.

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dEXL(x, mu=0.5, sigma=0.5), 
      from=0, to=5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dEXL(x, mu=1, sigma=0.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dEXL(x, mu=1.5, sigma=0.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=0.5", 
                            "mu=1.0, sigma=0.5",
                            "mu=1.5, sigma=0.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)


curve(dEXL(x, mu=0.5, sigma=1), 
      from=0, to=5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dEXL(x, mu=1, sigma=1),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dEXL(x, mu=1.5, sigma=1),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=1", 
                            "mu=1.0, sigma=1",
                            "mu=1.5, sigma=1"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)


curve(dEXL(x, mu=0.5, sigma=1.5), 
      from=0., to=8,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dEXL(x, mu=1, sigma=1.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dEXL(x, mu=1.5, sigma=1.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=1.5", 
                            "mu=1.0, sigma=1.5",
                            "mu=1.5, sigma=1.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)

# Example 2
# Checking if the cumulative curves converge to 1
curve(pEXL(x, mu=0.5, sigma=0.5), 
      from=0, to=5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="f(x)")
curve(pEXL(x, mu=1, sigma=0.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(pEXL(x, mu=1.5, sigma=0.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=0.5, sigma=0.5", 
                               "mu=1.0, sigma=0.5",
                               "mu=1.5, sigma=0.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.5)


# Example 3
# The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qEXL(p, mu=2.3, sigma=1.7), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pEXL(x, mu=2.3, sigma=1.7), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

# Comparing quantile, density and cumulative
p <- c(0.25, 0.5, 0.75)
quantile <- qEXL(p=p, mu=2.3, sigma=1.7) 

for(i in quantile){
  print(integrate(dEXL, lower=0, upper=i, mu=2.3, sigma=1.7))
}

pEXL(q=quantile, mu=2.3, sigma=1.7)

# Example 4
# The random function
x <- rEXL(n=10000, mu=1.5, sigma=2.5)
hist(x, freq=FALSE)
curve(dEXL(x, mu=1.5, sigma=2.5), from=0, to=20, 
      add=TRUE, col="tomato", lwd=2)

# Example 5
# The Hazard function
curve(hEXL(x, mu=1.5, sigma=2), from=0.001, to=4,
      col="tomato", ylab="Hazard function", las=1)