DiscreteDists (version 1.1.1)

DMOLBE: The DMOLBE family

Description

The function DMOLBE() defines the Discrete Marshall-Olkin Length Biased Exponential distribution a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

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

Value

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

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.

Author

Olga Usuga,

Details

The DMOLBE distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and mass function given by

\(f(x | \mu, \sigma) = \frac{\sigma ((1+x/\mu)\exp(-x/\mu)-(1+(x+1)/\mu)\exp(-(x+1)/\mu))}{(1-(1-\sigma)(1+x/\mu)\exp(-x/\mu)) ((1-(1-\sigma)(1+(x+1)/\mu)\exp(-(x+1)/\mu))}\)

with \(\mu > 0\) and \(\sigma > 0\)

References

Aljohani, H. M., Ahsan-ul-Haq, M., Zafar, J., Almetwally, E. M., Alghamdi, A. S., Hussam, E., & Muse, A. H. (2023). Analysis of Covid-19 data using discrete Marshall–Olkinin length biased exponential: Bayesian and frequentist approach. Scientific Reports, 13(1), 12243.

See Also

dDMOLBE.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(1234)
y <- rDMOLBE(n=100, mu=10, sigma=7)

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

# A function to simulate a data set with Y ~ DMOLBE
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu    <- exp(1.21 - 3 * x1) # 0.75 approximately
  sigma <- exp(1.26 - 2 * x2) # 1.30 approximately
  y <- rDMOLBE(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=200)

# Fitting the model
mod2 <- NULL
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=DMOLBE, data=dat,
                 control=gamlss.control(n.cyc=500, trace=FALSE))

summary(mod2)


# Example 3
# Data Set I (death due to coronavirus in China). The first data set is the number
# of deaths due to coronavirus in China from 23 January to 28 March.
# The data sets used in the paper was collected from 2020 year. The data set
# is reported in https://www.worldometers.info/coronavirus/country/china/.
# The data are:

y <- c(8, 16, 15, 24, 26, 26, 38, 43, 46, 45, 57, 64, 65, 73, 73, 86, 89, 97,
       108, 97, 146, 121, 143, 142, 105, 98, 136, 114, 118, 109, 97, 150, 71,
       52, 29, 44, 47, 35, 42, 31, 38, 31, 30, 28, 27, 22, 17, 22, 11, 7,
       13, 10, 14, 13, 11, 8, 3, 7, 6, 9, 7, 4, 6, 5, 3, 5)

# Fitting the model
mod3 <- gamlss(y~1, sigma.fo=~1, family=DMOLBE,
               control=gamlss.control(n.cyc=500, trace=FALSE))

summary(mod3)

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

# Example 4
# Data Set II (daily death due to coronavirus in Pakistan). The second data
# set is the daily deaths due to coronavirus in Pakistan from 18 March
# to 30 June. The data sets used in the paper was collected from 2020 year.
# The data is reported in
# https://www.worldometers.info/coronavirus/country/Pakistan.
# The data are:

y <- c(1, 6, 6, 4, 4, 4, 1, 20, 5, 2, 3, 15, 17, 7, 8, 25, 8, 25, 11,
       25, 16, 16, 12, 11, 20, 31, 42, 32, 23, 17, 19, 38, 50, 21, 14,
       37, 23, 47, 31, 24, 9, 64, 39, 30, 36, 46, 32, 50, 34, 32, 34,
       30, 28, 35, 57, 78, 88, 60, 78, 67, 82, 68, 97, 67, 65, 105,
       83, 101, 107, 88, 178, 110, 136, 118, 136, 153, 119, 89, 105,
       60, 148, 59, 73, 83, 49, 137, 91)

# Fitting the model
mod4 <- gamlss(y~1, sigma.fo=~1, family=DMOLBE,
               control=gamlss.control(n.cyc=500, trace=FALSE))

summary(mod4)

# Extracting the fitted values for mu and sigma
# using the inverse link function
mu_hat <- exp(coef(mod4, what="mu"))
mu_hat
sigma_hat <- exp(coef(mod4, what="sigma"))
sigma_hat