RelDists (version 1.0.2)

GLIN: The Generalized Lindley Type II family

Description

The Generalized Lindley Type II (GLIN) family for fitting positive continuous lifetime data within the GAMLSS framework.

Usage

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

Value

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

Arguments

mu.link

defines the mu.link, with "log" link as the default for the mu parameter (mu > 0).

sigma.link

defines the sigma.link, with "log" link as the default for the sigma parameter (sigma > 0).

Author

Sofia Cadavid Rueda,

Details

The Generalized Lindley Type II distribution with parameters mu and sigma has probability density function

\( f(x|\mu,\sigma)= \frac{\mu^2}{\mu+1} \left( 1+\frac{\mu^{\sigma-2}x^{\sigma-1}} {\Gamma(\sigma)} \right) e^{-\mu x}, \)

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

The distribution is a two-parameter extension of the classical Lindley distribution and belongs to the class of finite mixtures involving exponential and gamma components. It provides additional flexibility for modeling positively skewed lifetime data.

The original parameters of the distribution are denoted by \(\theta\) and \(\alpha\). In the GAMLSS implementation, they are re-parameterized as \(\mu=\theta\) and \(\sigma=\alpha\).

The \(r\)-th raw moment is given by

\( E(X^r)= \frac{1} {\mu^r(\mu+1)} \left[ \mu\Gamma(r+1) + \frac{\Gamma(r+\sigma)} {\Gamma(\sigma)} \right]. \)

In particular, the mean is

\( E(X)= \frac{\mu+\sigma} {\mu(\mu+1)}. \)

and the variance is

\( Var(X)= \frac{\mu^2 + \mu(\sigma^2 - \sigma + 2) + \sigma} {\mu^2(\mu + 1)^2}. \)

The GLIN distribution has been proposed for modeling lifetime and survival data and has shown greater flexibility than the classical Lindley and Exponential distributions in several applications.

References

Ekhosuehi, N., Opone, F., & Odobaire, F. (2018). A New Generalized Two Parameter Lindley Distribution. Journal of the Nigerian Statistical Association, 30, 547-566.

See Also

dGLIN

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(1234)
y <- rGLIN(n=500, mu=0.75, sigma=1.3)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=GLIN)

# 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 as GLIN
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.45 - 3 * x1)   # Approx 0.95
  sigma <- exp(2 - 1.5 * x2) # Approx 3.50
  y <- rGLIN(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}
set.seed(1234)
dat <- gendat(n=1000)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=GLIN, data=dat,
               control=gamlss.control(n.cyc=50, trace=FALSE))

summary(mod2)

# Example 3
# Remission times (in months) of 128 bladder cancer patients
# Taken from Lee and Wang (2003)
# Ekhosuehi et al. (2018) Table 4

y <- c(0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63,
       0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40,
       2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50,
       2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51,
       2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81,
       2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64,
       3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69,
       4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69,
       4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75,
       4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33,
       5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62,
       7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93,
       11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13,
       1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31,
       4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76,
       12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69)

require(gamlss)
mod3 <- gamlss(y ~ 1, sigma.fo = ~1, family = GLIN)

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

# Comparing the empirical histogram with the estimated density
hist(y, breaks=15, freq=FALSE,
     xlab="y", col="lightblue", border="white")
curve(dGLIN(x, mu=0.1247079, sigma=1.189219),
      add=TRUE, col="red", lwd=2)