The New Exponentiated Exponential distribution

Density, distribution function, quantile function, random generation and hazard function for the two-parameter New Exponentiated Exponential with parameters mu and sigma.

Usage

dNEE(x, mu = 1, sigma = 1, log = FALSE)

pNEE(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qNEE(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rNEE(n = 1, mu = 1, sigma = 1)

hNEE(x, mu, sigma, log = FALSE)

Arguments

x, q: vector of quantiles.
mu: parameter.
sigma: parameter.
log, log.p: logical; if TRUE, probabilities p are given as log(p).
lower.tail: logical; if TRUE (default), probabilities are \(P[X <= x]\), otherwise, \(P[X > x]\).
p: vector of probabilities.
n: number of observations.

Value

dNEE gives the density, pNEE gives the distribution function, qNEE gives the quantile function, rNEE generates random deviates and hNEE gives the hazard 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.

Author

Juliana Garcia, juliana.garciav@udea.edu.co

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dNEE(x, mu=0.2, sigma=0.3),
      from=0, to=8, col="cadetblue3", las=1, ylab="f(x)")

curve(dNEE(x, mu=1, sigma=4),
      add=TRUE, col= "purple")

curve(dNEE(x, mu=1.5, sigma=22),
      add=TRUE, col="goldenrod")

curve(dNEE(x, mu=0.5, sigma=2),
      add=TRUE, col="green3")

legend("topright", col=c("cadetblue3", "purple", "goldenrod", "green3"), lty=1, bty="n",
       legend=c("mu=0.2, sigma=0.3",
                "mu=1.0, sigma=4",
                "mu=1.5, sigma=22",
                "mu=0.5, sigma=2"))


# Example 2
# Checking if the cumulative curves converge to 1
curve(pNEE(x, mu=0.2, sigma=0.3), ylim=c(0, 1),
      from=0, to=8, col="cadetblue3", las=1, ylab="F(x)")

curve(pNEE(x, mu=1, sigma=4),
      add=TRUE, col= "purple")

curve(pNEE(x, mu=1.5, sigma=22),
      add=TRUE, col="goldenrod")

curve(pNEE(x, mu=0.5, sigma=2),
      add=TRUE, col="green3")

legend("bottomright", col=c("cadetblue3", "purple", "goldenrod", "green3"), lty=1, bty="n",
       legend=c("mu=0.2, sigma=0.3",
                "mu=1.0, sigma=4",
                "mu=1.5, sigma=22",
                "mu=0.5, sigma=2"))


# Example 3
# Checking the quantile function
mu <- 0.5
sigma <- 2
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qNEE(p, mu=mu, sigma=sigma), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pNEE(x, mu=mu, sigma=sigma), from=0, add=TRUE, col="red")


# Example 4
# Comparing the random generator output with
# the theoretical probabilities
mu <- 0.5
sigma <- 2
x <- rNEE(n=10000, mu=mu, sigma=sigma)
hist(x, freq=FALSE)
curve(dNEE(x, mu=mu, sigma=sigma), col="tomato", add=TRUE)