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")defines the mu.link, with "log" link as the default for the mu parameter.
defines the sigma.link, with "log" link as the default for the sigma.
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 AA, Hegazy MA, AL-Dayian GR, Abd EL-Kader RE (2022). “A Discrete Analog of the Inverted Kumaraswamy Distribution: Properties and Estimation with Application to COVID-19 Data.” Pakistan Journal of Statistics & Operation Research, 18(1).
# 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=150)
# 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) 1.0937 0.4966 2.202 0.0292 *
#> x1 -2.2099 0.9829 -2.248 0.0260 *
#> ---
#> 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) 3.4109 0.8629 3.953 0.000119 ***
#> x2 -5.4919 1.7364 -3.163 0.001896 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> ------------------------------------------------------------------
#> No. of observations in the fit: 150
#> Degrees of Freedom for the fit: 4
#> Residual Deg. of Freedom: 146
#> at cycle: 11
#>
#> Global Deviance: 851.1499
#> AIC: 859.1499
#> SBC: 871.1924
#> ******************************************************************