dDEOWE <- function(x, a, b, c, d){
exp(-((abs(x - c) / a) ^ b + abs(x + c) / d) ^ 2)
}New Distribution
DEOWE, New Discrete Distribution
Discrete distribution in statistics is a probability distribution that calculates the likelihood of a particular discrete, finite outcome. In simple words, the discrete probability distribution helps find the chances of the occurrence of a certain event expressed in terms of positive, non-decimal, or whole numbers as opposed to a continuous distribution. Discrete distribution is a very important statistical tool with diverse applications in economics, finance, and science. For example, it helps find the probability of an outcome and make predictions related to the stock market and the economy.
The Weibull distribution was introduced by the Swedish physicist Weibull. It has been used in many different fields like material science, engineering, physics, chemistry, meteorology, medicine, pharmacy, economics and business, quality control, biology, geology, and geography.
The Discrete Extended Odd Exponential Weibull Exponential (DEOWE) distribution is a discrete probability distribution that is used to model discrete data in various areas such as engineering, biology and economics. It is a generalization of the Poisson distribution, negative binomial distribution, and hypergeometric distribution.
The DEOWE distribution has applications in fields such as reliability analysis, queueing theory, and actuarial science. It provides a flexible and versatile model for the analysis of discrete data, especially when the data exhibit over- or under-dispersion compared to the Poisson distribution.
Functions
The probability mass function (PMF) of the DEOWE distribution is given by:
P(X =k) = [α(β + 1)(k - 1)!] / [(α + β + 1)(α + k - 1)!(β + k - 1)!], where k = 1, 2, 3, .…
Here, α and β are positive shape parameters of the distribution. The DEOWE distribution is said to be a discrete analogue of the Odd Weibull Exponential Weibull (OWE) distribution.
The mean and variance of the DEOWE distribution are:
- E(X) = α / (β + 1)
- Var(X) = [α(α + β)] / [(β + 1)²(β + 2)].
The DEOWE distribution also has a moment generating function (GMF), which is given by:
M(t) = [α + β + e^t(α + β)] / [(β + 1) + e^t(α + 1)].
The DEOWE distribution has applications in fields such as reliability analysis, queuing theory and actuarial science. In addition, various parameter estimation techniques and goodness-of-fit tests have been developed for this distribution over the years.
Let’s see an example in R of how to use the distribution. First we will create the distribution function:
Graphing it:
curve(dDEOWE(x, a = 1, b = 2, c = 3, d = 4), from = 0, to = 10,
main = "Probability density function DEOWE",
ylab = "Probability density", xlab = "Values of x")
Another example:
In this example, the deowe() function takes as input a vector of x values and the form parameter alpha.
deowe <- function(x, alpha) {
beta(alpha, 1/2) * alpha * x^(alpha-1) * exp(-x^alpha/2)
}A vector of x-values is created using the seq() function to generate values between 0 and 5 with a length of 100.
x <- seq(0, 5, length.out = 100)Finally, the DEOWE distribution function is plotted for three different values of alpha (1, 2, 3).
plot(x, deowe(x, 1), type = "l", col = "red", xlab = "x", ylab = "Density")
lines(x, deowe(x, 2), col = "blue")
lines(x, deowe(x, 3), col = "green")
legend("topright", legend = c("alpha = 1", "alpha = 2", "alpha = 3"), col = c("red", "blue", "green"), lty = 1)
Now let’s look at various forms of the distribution for alpha values of 0.2, 0.5, 0.8, 1, 1.5, 2, 4, 6, 6.5:
Generated from random numbers
We are going to implement a number generator for the Discrete Extended Odd Exponential Weibull Exponential (DEOWE):
n <- 1000000
alpha <- 1.5
set.seed(128)
u <- runif(n)
x <- (-log(1 - u^(1/alpha)))^(1/2)
hist(x, breaks = 40, prob = TRUE, col = "blue", xlab = "x", ylab = "Density", main = "Distribution DEOWE")
Comparison between the theorical and simulated:
n <- 1000000
alpha <- 1.5
set.seed(128)
u <- runif(n)
x <- (-log(1 - u^(1/alpha)))^(1/2)
f_deowe <- function(x, alpha) {
2 * alpha * x^(2*alpha-1) * exp(-x^(2*alpha))
}
hist(x, breaks = 40, prob = TRUE, col = "blue", xlab = "x", ylab = "Density", main = "Distribution DEOWE")
curve(f_deowe(x, alpha), add = TRUE, col = "red", lwd = 2)
legend("topright", c("simulated", "theorical"), fill = c("blue", "red"))