A New Probability Model: Statistical Properties, Estimation, Simulation, and Application

Osama M and Burki I

Published on: 2025-10-15

Abstract

This paper introduces a new probability distribution called the New Modified Exponential Gumbel (NMEG) distribution. The mathematical properties of the proposed distribution are thoroughly investigated, including its quantile function, moments, moment generating function, and order statistics. The maximum likelihood estimation (MLE) method is employed to estimate the parameters of the new model. A simulation study is conducted to assess the performance and consistency of the parameter estimates. The efficiency and applicability of the new model are further demonstrated using a real data set. The results show that the proposed model provides a better fit and demonstrates superiority over several well-known existing models.

Keywords

Gumbel distribution; Maximum likelihood estimation; Moments; Order statistics; Goodness of fit criteria

Introduction

Probability distributions are fundamental concepts in statistics and probability that describe how the values of random variables are distributed. They provide a mathematical framework for quantifying uncertainty and making predictions based on real data. Probability distributions specify the likelihood of each possible outcome in a random experiment and are crucial in fields such as mathematics, statistics, data science, and economics and engineering. Weibull, Rayleigh, and exponential distributions are among the most commonly used models in practical data analysis due to their effectiveness in representing real-life phenomena.

The Gumbel distribution, also known as the Extreme Value Type I distribution, is widely used in statistics to model the distribution of the minimum number of sample observations. It is especially useful in extreme value theory, which focuses on the behavior of the tails of a distribution. The probability density function (pdf) and cumulative density function (cdf) of Gumbel distribution is given as;
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