DIKUM function

The discrete Inverted Kumaraswamy family

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")

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.

Returns

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

Details

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.

Examples

# 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'))
exp(coef(mod1, what='sigma'))

# 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)

References

\insertRef EL_Helbawy2022DiscreteDists

See Also

dDIKUM .

Author(s)

Daniel Felipe Villa Rengifo, dvilla@unal.edu.co