The function DBH() defines the Discrete Burr Hatke distribution, one-parameter
discrete distribution, for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
DBH(mu.link = "logit")defines the mu.link, with "logit" link as the default for the mu parameter. Other links are "probit" and "cloglog"'(complementary log-log)
Returns a gamlss.family object which can be used
to fit a Discrete Burr-Hatke distribution
in the gamlss() function.
The Discrete Burr-Hatke distribution with parameters \(\mu\) has a support 0, 1, 2, ... and density given by
\(f(x | \mu) = (\frac{1}{x+1}-\frac{\mu}{x+2})\mu^{x}\)
The pmf is log-convex for all values of \(0 < \mu < 1\), where \(\frac{f(x+1;\mu)}{f(x;\mu)}\) is an increasing function in \(x\) for all values of the parameter \(\mu\).
Note: in this implementation we changed the original parameters \(\lambda\) for \(\mu\), we did it to implement this distribution within gamlss framework.
El-Morshedy M, Eliwa MS, Altun E (2020). “Discrete Burr-Hatke distribution with properties, estimation methods and regression model.” IEEE access, 8, 74359--74370.
dDBH.
# Example 1
# Generating some random values with
# known mu
y <- rDBH(n=1000, mu=0.74)
library(gamlss)
#> Loading required package: splines
#> Loading required package: gamlss.data
#>
#> Attaching package: 'gamlss.data'
#> The following object is masked from 'package:datasets':
#>
#> sleep
#> Loading required package: gamlss.dist
#> Loading required package: nlme
#> Loading required package: parallel
#> ********** GAMLSS Version 5.4-22 **********
#> For more on GAMLSS look at https://www.gamlss.com/
#> Type gamlssNews() to see new features/changes/bug fixes.
mod1 <- gamlss(y~1, family=DBH,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse logit function
inv_logit <- function(x) exp(x) / (1+exp(x))
inv_logit(coef(mod1, parameter="mu"))
#> (Intercept)
#> 0.7572509
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ DBH
gendat <- function(n) {
x1 <- runif(n)
mu <- inv_logit(-3 + 5 * x1)
y <- rDBH(n=n, mu=mu)
data.frame(y=y, x1=x1)
}
datos <- gendat(n=150)
mod2 <- NULL
mod2 <- gamlss(y~x1, family=DBH, data=datos,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family: c("DBH", "Burr Hatke")
#>
#> Call:
#> gamlss(formula = y ~ x1, family = DBH, data = datos, control = gamlss.control(n.cyc = 500,
#> trace = FALSE))
#>
#> Fitting method: RS()
#>
#> ------------------------------------------------------------------
#> Mu link function: logit
#> Mu Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -3.3473 0.6124 -5.466 1.91e-07 ***
#> x1 5.4145 0.8583 6.309 3.08e-09 ***
#> ---
#> 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: 2
#> Residual Deg. of Freedom: 148
#> at cycle: 2
#>
#> Global Deviance: 192.5091
#> AIC: 196.5091
#> SBC: 202.5304
#> ******************************************************************
# Example 3
# number of carious teeth among the four deciduous molars.
# Taken from EL-MORSHEDY (2020) page 74364.
y <- rep(0:4, times=c(64, 17, 10, 6, 3))
mod3 <- gamlss(y~1, family=DBH,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod3, what="mu"))
#> (Intercept)
#> 0.6712271