The function NEE() defines the New Exponentiated Exponential distribution, a two parameter
distribution, for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
Value
Returns a gamlss.family object which can be used to fit a
NEE distribution in the gamlss() function.
Details
The New Exponentiated Exponential distribution with parameters mu
and sigma has density given by
\(f(x | \mu, \sigma) = \log(2^\sigma) \mu \exp(-\mu x) (1-\exp(-\mu x))^{\sigma-1} 2^{(1-\exp(-\mu x))^\sigma}, \)
for \(x>0\), \(\mu>0\) and \(\sigma>0\).
Note: In this implementation we changed the original parameters \(\theta\) for \(\mu\) and \(\alpha\) for \(\sigma\), we did it to implement this distribution within gamlss framework.
References
Hassan, Anwar, I. H. Dar, and M. A. Lone. "A New Class of Probability Distributions With An Application to Engineering Data." Pakistan Journal of Statistics and Operation Research 20.2 (2024): 217-231.
Examples
# Example 1
# Generating some random values with
# known mu and sigma
y <- rNEE(n=500, mu=2.5, sigma=3.5)
# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=NEE,
control=gamlss.control(n.cyc=5000, trace=TRUE))
#> Error in estim_mu_sigma_NEE(y): could not find function "estim_mu_sigma_NEE"
# Extracting the fitted values for mu, sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
#> Error: object 'mod1' not found
exp(coef(mod1, what="sigma"))
#> Error: object 'mod1' not found
# Example 2
# Generating random values under some model
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(-0.2 + 1.5 * x1)
sigma <- exp(1 - 0.7 * x2)
y <- rNEE(n=n, mu, sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(123)
datos <- gendat(n=500)
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=NEE, data=datos,
control=gamlss.control(n.cyc=5000, trace=TRUE))
#> Error in estim_mu_sigma_NEE(y): could not find function "estim_mu_sigma_NEE"
summary(mod2)
#> Error: object 'mod2' not found
# Example 3 --------------------------------------------------
# Obtained from Hassan (2024) page 226
# The data set consists of 63 observations of the gauge lengths of 10mm.
y <- c(1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397,
2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614,
2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917,
2.928, 2.937, 2.937, 2.977, 2.996, 3.030, 3.125, 3.139, 3.145,
3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346,
3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628,
3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020)
mod3 <- gamlss(y~1, family=NEE)
#> Error in estim_mu_sigma_NEE(y): could not find function "estim_mu_sigma_NEE"
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod3, what="mu"))
#> Error: object 'mod3' not found
exp(coef(mod3, what="sigma"))
#> Error: object 'mod3' not found
# Hist and estimated pdf
hist(y, freq=FALSE, ylim=c(0, 0.7))
curve(dNEE(x, mu=2.076862, sigma=255.2289),
add=TRUE, col="tomato", lwd=2)
# Empirical cdf and estimated ecdf
plot(ecdf(y))
curve(pNEE(x, mu=2.076862, sigma=255.2289),
add=TRUE, col="tomato", lwd=2)
# QQplot
qqplot(y, rNEE(n=length(y), mu=2.076862, sigma=255.2289), col="tomato")
qqline(y, distribution=function(p) qNEE(p, mu=2.076862, sigma=255.2289))
# Example 4 --------------------------------------------------
# Obtained from Hassan (2024) page 226
# The dataset was reported by Bader and Priest (1982) on failure
# stresses (in GPa) of 65 single carbon fibers of lengths 50 mm
y <- c(0.564, 0.729, 0.802, 0.95, 1.053, 1.111, 1.115, 1.194, 1.208,
1.216, 1.247, 1.256, 1.271, 1.277, 1.305, 1.313, 1.348,
1.39, 1.429, 1.474, 1.49, 1.503, 1.52, 1.522, 1.524, 1.551,
1.551, 1.609, 1.632, 1.632, 1.676, 1.684, 1.685, 1.728, 1.74,
1.761, 1.764, 1.785, 1.804, 1.816, 1.824, 1.836, 1.879, 1.883,
1.892, 1.898, 1.934, 1.947, 1.976, 2.02, 2.023, 2.05, 2.059,
2.068, 2.071, 2.098, 2.13, 2.204, 2.317, 2.334, 2.34, 2.346,
2.378, 2.483, 2.269)
mod4 <- gamlss(y~1, family=NEE)
#> Error in estim_mu_sigma_NEE(y): could not find function "estim_mu_sigma_NEE"
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod4, what="mu"))
#> Error: object 'mod4' not found
exp(coef(mod4, what="sigma"))
#> Error: object 'mod4' not found
hist(y, freq=FALSE)
curve(dNEE(x, mu=2.400515, sigma=25.15236),
add=TRUE, col="tomato", lwd=2)
# Empirical cdf and estimated ecdf
plot(ecdf(y))
curve(pNEE(x, mu=2.400515, sigma=25.15236),
add=TRUE, col="tomato", lwd=2)
# QQplot
qqplot(y, rNEE(n=length(y), mu=2.400515, sigma=25.15236), col="tomato")
qqline(y, distribution=function(p) qNEE(p, mu=2.400515, sigma=25.15236))
# Example 5 -------------------------------------------------------------------
# 69 Observations of the gauge lengths of 20m.
y <- c(1.312,1.314,1.479,1.552,1.700,1.803,1.861,1.865,1.944,1.958,1.966,1.997,
2.006,2.021,2.027,2.055, 2.063,2.098,2.140,2.179,2.224,2.240,2.253,2.270,
2.272,2.274,2.301,2.301,2.359,2.382,2.382,2.426, 2.434,2.435,2.478,2.490,
2.511,2.514,2.535,2.554,2.566,2.570,2.586,2.629,2.633,2.642,2.648,2.684,
2.697,2.726,2.770,2.773,2.800,2.809,2.818,2.821,2.848,2.880,2.954,3.012,
3.067,3.084,3.090,3.096, 3.128,3.233,3.433,3.585,3.585)
mod5 <- gamlss(y~1, sigma.fo=~1, family = NEE)
#> Error in estim_mu_sigma_NEE(y): could not find function "estim_mu_sigma_NEE"
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod5, what="mu"))
#> Error: object 'mod5' not found
exp(coef(mod5, what="sigma"))
#> Error: object 'mod5' not found
hist(y, freq=FALSE)
curve(dNEE(x, mu=2.197771, sigma=100.8888), add=TRUE,
col="tomato", lwd=2)
# Empirical cdf and estimated ecdf
plot(ecdf(y))
curve(pNEE(x, mu=2.197771, sigma=100.8888), add=TRUE,
col="tomato", lwd=2)
# QQplot
qqplot(y, rNEE(n=length(y), mu=2.197771, sigma=100.8888), col="tomato")
qqline(y, distribution=function(p) qNEE(p, mu=2.197771, sigma=100.8888))
