Kth-order equilibrium Weibull distribution: properties, simulation, and its applications

Abstract

A generalization of some well-known probability distributions (viz., Weibull, exponential, gamma, Maxwell, and Chi-square) is introduced using the concept of weighted probability distributions. The introduced model is named Kth-order equilibrium Weibull distribution (KEWD). Various properties of the new distribution are studied in detail. One of the important properties of KEWD is that its hazard rate shows increasing, decreasing, and constant behavior. It is also shown that the new model belongs to the log exponential family. Various ordering relations of the KEWD are studied in comparison with the baseline model, i.e. Weibull distribution. Parameters are estimated using the concept of the maximum likelihood estimation technique. Using the Anderson–Darling test statistic, a simulation study is carried out to analyze the asymptotic normality behavior of maximum likelihood estimators. The behaviors of bias and mean square error are observed with the increase in sample size. The applications of new distribution are illustrated through its fitting to some incorporated real-life data sets. Finally, a comparison is made between KEWD, its sub-models, and some other introduced extensions of Weibull distribution in terms of fitting using the Akaike information criterion (AIC).

Keywords:

• Equilibrium distribution
• Aging properties
• Maximum likelihood estimation
• Anderson–Darling test
• Asymptotic normality
• Akaike information criterion

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 62E15
• 62F10
• 62G30

Acknowledgments

The authors would like to thank the Senior Editor, for his invaluable contributions and constant encouragement. They would also like to express their most profound appreciation to the anonymous referee, whose helpful assistance, constructive feedback, and insightful suggestions have greatly enhanced the quality and clarity of the article.