The function DIKUM() defines the discrete Inverted Kumaraswamy distribution, a two parameter
distribution, for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
DIKUM(mu.link = "log", sigma.link = "log")Returns a gamlss.family object which can be used
to fit a discrete Inverted Kumaraswamy distribution
in the gamlss() function.
The discrete Inverted Kumaraswamy distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and density given by
\(f(x | \mu, \sigma) = (1-(2+x)^{-\mu})^{\sigma}-(1-(1+x)^{-\mu})^{\sigma}\)
with \(\mu > 0\) and \(\sigma > 0\).
Note: in this implementation we changed the original parameters \(\alpha\) and \(\beta\) for \(\mu\) and \(\sigma\) respectively, we did it to implement this distribution within gamlss framework.
El-Helbawy, A. A., Hegazy, M. A., Al-Dayian, G. R., & Abd EL-Kader, R. E. (2022). A discrete analog of the inverted Kumaraswamy distribution: properties and estimation with application to COVID-19 data. Pakistan Journal of Statistics and Operation Research, 18(1), 297-328.
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(150)
y <- rDIKUM(1000, mu=1, sigma=5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y ~ 1, sigma.fo = ~1, family=DIKUM,
control = gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
#> (Intercept)
#> 0.9977962
exp(coef(mod1, what="sigma"))
#> (Intercept)
#> 4.955013
# Example 2
# Generating random values under some model
library(gamlss)
# A function to simulate a data set with Y ~ DIKUM
gendat <- function(n) {
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(1.21 - 3 * x1) # 0.75 approximately
sigma <- exp(1.26 - 2 * x2) # 1.30 approximately
y <- rDIKUM(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
dat <- gendat(n=1000)
# Fitting the model
mod2 <- gamlss(y ~ x1, sigma.fo = ~x2, family = "DIKUM", data=dat,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family: c("DIKUM", "discrete-Inverted-Kumaraswamy")
#>
#> Call: gamlss(formula = y ~ x1, sigma.formula = ~x2, family = "DIKUM",
#> data = dat, control = gamlss.control(n.cyc = 500, trace = FALSE))
#>
#> Fitting method: RS()
#>
#> ------------------------------------------------------------------
#> Mu link function: log
#> Mu Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1.1570 0.2271 5.096 4.15e-07 ***
#> x1 -2.8971 0.4560 -6.353 3.20e-10 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ------------------------------------------------------------------
#> Sigma link function: log
#> Sigma Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1.7099 0.3383 5.055 5.13e-07 ***
#> x2 -2.7532 0.6734 -4.089 4.69e-05 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ------------------------------------------------------------------
#> No. of observations in the fit: 1000
#> Degrees of Freedom for the fit: 4
#> Residual Deg. of Freedom: 996
#> at cycle: 13
#>
#> Global Deviance: 6219.8
#> AIC: 6227.8
#> SBC: 6247.431
#> ******************************************************************