Two-parameter family of Unit distributions: theory, statistical inference, and reliability applications

Document Type

Article

Publication Date

Summer 9-9-2026

Abstract

This paper develops a two-parameter family of unit distributions (TPFUD) by applying the standard unit transformation to a positive-support parent family to model data on the open unit interval (0,1), such as proportions, rates, percentages, and transformed lifetime observations. The proposed family admits tractable forms for the density, distribution, survival, hazard, cumulative hazard, reversed hazard, and odds functions, and it provides a unified framework for several Lindley-type unit models. Its main mathematical properties are derived, including its 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 unit Shanker distribution (UShD), is then examined in detail. For this model, maximum likelihood, the maximum product of spacings, and Bayesian estimation methods are developed for the unknown parameter and selected reliability functions. The existence and uniqueness of the maximum likelihood estimator are also established. A Monte Carlo simulation study shows that the estimators improve as the sample size increases, with informative Bayesian estimation giving the best finite-sample performance. Two real-data applications further demonstrate that the UShD provides competitive and flexible fits compared with existing unit distributions.

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