DiscreteDists (version 1.1.1)

DLD: The Discrete Lindley family

Description

The function DLD() defines the Discrete Lindley distribution a single parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

DLD(mu.link = "log")

Value

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

Arguments

mu.link

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

Author

Yojan Andrés Alcaraz Pérez,

Details

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

\(f(x | \mu) = \frac{e^{-\mu x}}{1 + \mu} (\mu(1 - 2e^{-\mu}) + (1- e^{-\mu})(1+\mu x))\)

The parameter \(\mu\) can be interpreted as a strict upper bound on the failure rate function

The conventional discrete distributions (such as geometric, Poisson, etc.) are not suitable for various scenarios like reliability, failure times, and counts. Consequently, alternative discrete distributions have been created by adapting well-known continuous models for reliability and failure times. Among these, the discrete Weibull distribution stands out as the most widely used. But models like these require two parameters and not many of the known discrete distributions can provide accurate models for both times and counts, which the Discrete Lindley distribution does.

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

References

Bakouch, H. S., Jazi, M. A., & Nadarajah, S. (2014). A new discrete distribution. Statistics, 48(1), 200-240.

See Also

dDLD.

Examples

# Example 1
# Generating some random values with
# known mu
y <- rDLD(n=100, mu=0.3)

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

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

# Example 2
# Generating random values under some model

# A function to simulate a data set with Y ~ DLD
gendat <- function(n) {
  x1 <- runif(n)
  mu    <- exp(2 - 4 * x1)
  y <- rDLD(n=n, mu=mu)
  data.frame(y=y, x1=x1)
}

set.seed(1235)
datos <- gendat(n=150)

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

summary(mod2)

# Example 3
# Survival times in days of 72 guinea pigs.
# Taken from Bakouch et al (2014) page 26.

y <- c(12, 15, 22, 24, 24, 32, 32, 33, 34, 38, 38, 43, 44, 48,
       52, 53, 54, 54, 55, 56, 57, 58, 58, 59, 60, 60, 60, 60,
       61, 62, 63, 65, 65, 67, 68, 70, 70, 72, 73, 75, 76, 76,
       81, 83, 84, 85, 87, 91, 95, 96, 98, 99, 109, 110, 121,
       127, 129, 131, 143, 146, 146, 175, 175, 211, 233, 258,
       258, 263, 297, 341, 341, 376)

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

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


# Example 4
# Remission times in weeks for 20 leukaemia patients.
# Taken from Bakouch et al (2014) page 26.

y <- c(1, 3, 3, 6, 7, 7, 10, 12, 14, 15, 18, 19,
       22, 26, 28, 29, 34, 40, 48, 49)

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

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

# Example 5
# Numbers of fires in Greece for the period from 1
# July 1998 to 31 August of the same year .
# Taken from Bakouch et al (2014) page 26.

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

mod5 <- gamlss(y~1, family=DLD,
               control=gamlss.control(n.cyc=500, trace=FALSE))

# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod5, what="mu"))

# Example 6
# Final examination marks of space students in
# mathematics in the Indian Institute of Technology at Kanpur.
# Taken from Bakouch et al (2014) page 26.

y <- c(29, 25, 50, 15, 13, 27, 15, 18, 7, 7, 8, 19, 12,
       18, 5, 21, 15, 86, 21, 15, 14, 39, 15, 14, 70,
       44, 6, 23, 58, 19, 50, 23, 11, 6, 34, 18, 28, 34,
       12, 37, 4, 60, 20, 23, 40, 65, 19, 31)

mod6 <- gamlss(y~1, family=DLD,
               control=gamlss.control(n.cyc=500, trace=FALSE))

# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod6, what="mu"))