Theoretical and inferential aspects of a novel two-parameter log-family of distributions with applications

Abstract

This paper introduces a new two-parameter log-family of distributions for modeling data defined on the open unit interval (0, 1), such as proportions, rates, and percentage-based lifetime observations. The proposed family has tractable expressions for the density, distribution, survival, hazard, cumulative hazard, reversed hazard, and odds functions, and it provides a unified framework for several log-Lindley-type models. Its main mathematical properties are derived, including limiting behavior, quantile function, moments, incomplete moments, inequality measures, actuarial quantities, stress–strength reliability, order statistics, stochastic ordering, and entropy and extropy measures. A flexible one-parameter special case, called the log-linear-exponential distribution (LLED), is then examined in detail. For this model, maximum likelihood, maximum product of spacings, and Bayesian estimation methods are developed for the unknown parameter. Bayesian inference is implemented under an informative gamma prior and the non-informative scale-invariant prior using the Metropolis–Hastings algorithm. A Monte Carlo simulation study shows that the estimators improve as the sample size increases, with informative-prior Bayesian methods providing the best finite-sample performance. Three real-data applications further demonstrate that the LLED provides a competitive, parsimonious, and empirically adequate fit compared with several existing unit distributions.