Theoretical Properties of the Lindley Exponentiated Gumbel Distribution

This study introduces a new statistical distribution, named the Lindley Exponentiated Gumbel (LEGu) distribution, aiming to enhance the flexibility and adaptability of statistical models for various environmental datasets. The distribution is constructed by combining elements of the Lindley density function with its corresponding cumulative density function (cdf) and probability distribution function (pdf), offering increased flexibility and versatility for statistical modelling and analysis in various fields of research and application. To derive insights into the newly proposed distribution, the study investigates its structural properties and characteristics and presents expansions for its probability density and cumulative density functions using generalized binomial expansion. Several important representations, such as the survival function, hazard function, quantile function, probability weighted moment, moment generating function, and distribution of order statistics, are provided for the LEGu distribution. The method of estimation involves maximum likelihood estimation, with the sample log-likelihood function derived. Due to the complex nature of the likelihood function, numerical optimization techniques like the Newton-Raphson method are proposed for estimating the distribution parameters effectively. The proposed distribution's versatility and robustness, coupled with the use of maximum likelihood estimators, pave the way for more accurate and reliable interpretations of environmental data, leading to valuable insights and potential applications in diverse environmental research and decision-making processes.