Statistical Inference of Extropy Measures: A One-Parameter Weighted Distribution With Application to Environmental Data

In this paper, we introduce and study the length-biased Garhy distribution (LBGD), a novel one-parameter weighted extension of the Garhy distribution (GD) using the length-biased class of distributions. The LBGD offers enhanced flexibility for modeling positively skewed and heavy-tailed data commonly encountered in environmental analysis. Some important closed-form expressions for seven different extropy measures, such as classical extropy, weighted extropy, residual extropy, weighted residual extropy, past extropy, weighted past extropy, and cumulative residual extropy, are computed for the LBGD. In addition, several closed-form statistical properties, including mode, ordinary moments, mean, variance, moment generating function, and incomplete moments, are calculated. Parameter and extropy estimations are performed via maximum likelihood estimation methodology and are investigated. An extensive Monte Carlo simulation study evaluates the performance of the maximum likelihood estimator, as well as the estimators for all seven extropy measures, in terms of bias, mean squared error, root mean squared error, average interval length, and coverage probability. The importance, applicability, and superiority of the suggested LBGD are demonstrated through application to real environmental data. The LBGD consistently outperforms 15 well-known competing statistical distributions—such as LB weighted Ishita, Akash, Ishita, Pranav, Rama, half-logistic exponential, half-logistic Pareto, Xgamma, Chris-Jerry, Zeghdoudi, Shukla, Lindley, XLindley, Komal, and GD distributions—across several model selection criteria, goodness-of-fit tests, and extropy measure-based comparisons, confirming its potential as a valuable tool for analyzing positively skewed data.