DiscreteDists (version 1.1.1)

DsPA: Discrete power-Ailamujia distribution

Description

The function DsPA() defines the discrete power-Ailamujia distribution a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

DsPA(mu.link = "log", sigma.link = "logit")

Value

Returns a gamlss.family object which can be used to fit a discrete power-Ailamujia distribution in the gamlss() function.

Arguments

mu.link

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

sigma.link

defines the sigma.link, with "logit" link as the default for the sigma parameter. Other links are "probit" and "cloglog"'(complementary log-log).

Author

Maria Camila Mena Romana,

Details

The discrete power-Ailamujia distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and density given by

\(f(x | \mu, \sigma) = (\sigma^x)^\mu (1-x^\mu \log(\lambda)) - (\sigma^{(x+1)})^\mu (1-(x+1)^\mu \log(\lambda))\)

Note: in this implementation we changed the original parameters \(\beta\) and \(\lambda\) for \(\mu\) and \(\sigma\) respectively, we did it to implement this distribution within gamlss framework.

References

Alghamdi, A. S., Ahsan-ul-Haq, M., Babar, A., Aljohani, H. M., Afify, A. Z., & Cell, Q. E. (2022). The discrete power-Ailamujia distribution: properties, inference, and applications. AIMS Math, 7(5), 8344-8360.

See Also

dDsPA.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rDsPA(n=100, mu=1.2, sigma=0.5)

# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=DsPA,
               control=gamlss.control(n.cyc=500, trace=TRUE))

# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))

exp(coef(mod1, what="mu"))
inv_logit(coef(mod1, what="sigma"))

# Example 2
# Generating random values under some model

# A function to simulate a data set with Y ~ DsPA
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  x3 <- runif(n)
  x4 <- runif(n)
  mu    <- exp(1 + 1.2*x1 + 0.2*x2)
  sigma <- inv_logit(2 + 1.5*x3 + 1.5*x4)
  y <- rDsPA(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2, x3=x3, x4=x4)
}

set.seed(123)
datos <- gendat(n=100)

mod2 <- gamlss(y~x1+x2, sigma.fo=~x3+x4, family=DsPA, data=datos,
               control=gamlss.control(n.cyc=500, trace=TRUE))

summary(mod2)

# Example 3
# failure times for a sample of 15 electronic components in an acceleration life test
# Taken from
# Alghamdi et. al (202) page 8354.

y <- c(1.0, 5.0, 6.0, 11.0, 12.0, 19.0, 20.0, 22.0,
       23.0, 31.0, 37.0, 46.0, 54.0, 60.0, 66.0)

mod3 <- gamlss(y~1, family=DsPA,
               control=gamlss.control(n.cyc=500, trace=FALSE))

# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))

exp(coef(mod3, what="mu"))
inv_logit(coef(mod3, what="sigma"))

# Example 4
# number of fires in Greece from July 1, 1998 to August 31, 1998.
# Taken from
# Alghamdi et. al (202) page 8354.

y <- c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2,
       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,
       3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4,
       4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5,
       5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6,
       6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7,
       8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9,
       9, 9, 9, 9, 10, 10, 10, 11, 11,
       11, 11, 12, 12, 12, 12, 12, 12,
       15, 15, 15, 15, 16, 20, 43)

mod4 <- gamlss(y~1, family=DsPA,
               control=gamlss.control(n.cyc=500, trace=FALSE))

# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))

exp(coef(mod4, what="mu"))
inv_logit(coef(mod4, what="sigma"))