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

Hebatalla H. Mohammad; Moustafa N. Mousa; Khalaf S. Sultan; Mahmoud M. M. Mansour; Hebatalla H. Mohammad; Moustafa N. Mousa; Khalaf S. Sultan; Mahmoud M. M. Mansour

doi:10.3934/math.20261150

AIMS Mathematics

2026, Volume 11, Issue 9: 28887-28949. doi: 10.3934/math.20261150

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

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt
3.
Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt
4.
Department of Basic Science, Faculty of Engineering, The British University in Egypt, El Sherouk City, Cairo 11837, Egypt

Received: 30 April 2026 Revised: 24 August 2026 Accepted: 01 September 2026 Published: 09 September 2026
MSC : 60G35, 62F10, 62F15, 62N01, 62N02, 62N05

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.
- family of unit distributions,
- unit shanker distribution,
- reliability analysis,
- maximum likelihood,
- maximum product spacing,
- Bayesian estimation,
- Monte Carlo simulation
Citation: Hebatalla H. Mohammad, Moustafa N. Mousa, Khalaf S. Sultan, Mahmoud M. M. Mansour. Two-parameter family of Unit distributions: theory, statistical inference, and reliability applications[J]. AIMS Mathematics, 2026, 11(9): 28887-28949. doi: 10.3934/math.20261150

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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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