DiscreteDists (version 1.1.1)

DPERKS: The Discrete Perks family

Description

The function DPERKS() defines the Discrete Perks distribution a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

DPERKS(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.

Author

Veronica Seguro Varela,

Details

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

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

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

References

Tyagi, A., Choudhary, N., & Singh, B. (2020). A new discrete distribution: Theory and applications to discrete failure lifetime and count data. J. Appl. Probab. Statist, 15, 117-143.

See Also

dDPERKS.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(123)
y <- rDPERKS(n=1000, mu=2.5, sigma=0.4)

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

# 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 under some model

# A function to simulate a data set with Y ~ DPERKS
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu    <- exp(-1.6 + 5 * x1)
  sigma <- exp(1.7 - 5 * x2)
  y <- rDPERKS(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(12345)
datos <- gendat(n=1000)

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

summary(mod2)

# Example 3
# The dataset comes from Tyagi et al. (2020) page 21
# The dataset contains the number of outbreaks of strikes in the
# UK coal mining industries in four successive week periods
# in the year 1948-59.
y <- rep(0:4, times=c(46, 76, 24, 9, 1))

# Fitting the model
library(gamlss)
mod3 <- gamlss(y~1, family=DPERKS,
               control=gamlss.control(n.cyc=500, trace=TRUE))

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

# Example 4
# The dataset comes from Tyagi et al. (2020) page 21
# The dataset contains the number fishes caught
# in traps (David, 1971, pg. 168).
y <- rep(0:9, times=c(1, 2, 11, 20, 29, 23, 10, 3, 1, 0))

# Fitting the model
library(gamlss)
mod4 <- gamlss(y~1, family=DPERKS,
               control=gamlss.control(n.cyc=500, trace=TRUE))

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod4, what="mu"))
exp(coef(mod4, what="sigma"))

# Example 5
# The dataset comes from Tyagi et al. (2020) page 24
# This dataset consists of remission times in weeks
# for 20 leukemia patients randomly assigned to a certain treatment
y <- c(1, 3, 3, 6, 7, 7, 10, 12, 14, 15, 18,
       19, 22, 26, 28, 29, 34, 40, 48, 49)

# Fitting the model
library(gamlss)
mod4 <- gamlss(y~1, family=DPERKS)

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod4, what="mu"))
exp(coef(mod4, what="sigma"))