EXL: The exponentiated XLindley family

In ousuga/RelDists: Estimation for some Reliability Distributions

The exponentiated XLindley family

Description

The function EXL() defines The exponentiated XLindley, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

EXL(mu.link = "log", sigma.link = "log")

Arguments

mu.link	defines the mu.link, with "log" link as the default for the mu parameter.
sigma.link	defines the sigma.link, with "log" link as the default for the sigma.

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.

Value

Returns a gamlss.family object which can be used to fit a EXL distribution in the gamlss() function.

Author(s)

Manuel Gutierrez Tangarife, [e-mail]

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
# Generating some random values with
# known mu and sigma
y <- rEXL(n=300, mu=0.75, sigma=1.3)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=EXL,
              control=gamlss.control(n.cyc=5000, trace=FALSE))

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))


# Example 2
# Generating random values for a regression model

# A function to simulate a data set with Y ~ EXL
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.45 - 3 * x1)
  sigma <- exp(2 - 1.5 * x2)
  y <- rEXL(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2)
}

set.seed(1234)
dat <- gendat(n=100)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=EXL, data=dat,
               control=gamlss.control(n.cyc=5000, trace=FALSE))

summary(mod2)

# Example 3
# Mortality rate due to COVID-19 for 30 days (31st March to April 30, 2020)
# recorded for the Netherlands.
# Taken from Alomair et al. (2024) page 12.

x <- c(14.918, 10.656, 12.274, 10.289, 10.832, 7.099, 5.928, 13.211, 
       7.968, 7.584, 5.555, 6.027, 4.097, 3.611, 4.960, 7.498, 6.940, 
       5.307, 5.048, 2.857, 2.254, 5.431, 4.462, 3.883,
       3.461, 3.647, 1.974, 1.273, 1.416, 4.235)

mod3 <- gamlss(x~1, sigma.fo=~1, family=EXL,
               control=gamlss.control(n.cyc=5000, trace=FALSE))

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod3, what="mu"))
exp(coef(mod3, what="sigma"))

# Replicating figure 4 from Alomair et al. (2024)
# Hist and estimated pdf
hist(x, freq=FALSE)
curve(dEXL(x, mu=0.4089915, sigma=2.710467), add=TRUE, col="tomato", lwd=2)
# Empirical cdf and estimated ecdf
plot(ecdf(x))
curve(pEXL(x, mu=0.4089915, sigma=2.710467), add=TRUE, col="tomato", lwd=2)
# QQplot
qqplot(x, rEXL(n=30, mu=0.4089915, sigma=2.710467), col="tomato")
qqline(x, distribution=function(p) qEXL(p, mu=0.4089915, sigma=2.710467))


# Example 4
# Precipitation in inches
# Taken from Alomair et al. (2024) page 13.

# Manuel

ousuga/RelDists documentation built on March 29, 2025, 4:07 a.m.