Statistical Inference of the Mixed Exponential–Beta Distribution

Abstract

The classical exponential (Exp) distribution may have limited capability in describing asymmetric lifetime data and observations with different tail characteristics. In this paper, a new extended exponential model, called the Mixed Exponential–Beta (MEB) distribution, is proposed based on a scale-mixture representation involving exponential and beta distributions. The main statistical properties of the proposed distribution are investigated, including its probability density function (pdf), cumulative distribution function (cdf), reliability function, moments, skewness, and kurtosis. The additional shape parameters of the MEB distribution provide enhanced control over distributional shape, asymmetry, and tail behavior. The model parameters are estimated using the expectation–maximization (EM) algorithm based on the stochastic representation of the mixture model. The proposed EM-based maximum likelihood (ML) estimation method is further compared with the least-squares (LS) estimation method, and the results demonstrate its superior estimation accuracy. Finally, the applicability of the MEB distribution is illustrated through maintenance-time data, where the proposed model shows competitive fitting performance compared with several exponential-type distributions. The results indicate that the MEB distribution provides a flexible alternative for modeling positive lifetime data with different distributional characteristics. Moreover, the exponential distribution is obtained as a limiting case under appropriate parameter conditions.