Heterogeneous Mixture Distributions for Modeling Skewed Data in Engineering Applications: A Theoretical Approach

Mixture distribution models have become indispensable tools for characterizing complex, non-Gaussian data structures that frequently arise in engineering and reliability contexts. However, existing mixture frameworks predominantly assume homogeneous component families, typically Gaussian, which limits their capacity to represent data generated by fundamentally different physical mechanisms. This paper presents a formal theoretical framework for heterogeneous mixture distributions, in which component densities are drawn from distinct parametric families — specifically the Normal, Gamma, Weibull, and Lognormal distributions — with identifiability conditions formally established across the full family set. A unified parameter estimation strategy based on maximum likelihood and the Expectation-Maximization (EM) algorithm is developed, accommodating cross-family heterogeneity within a single iterative procedure. The framework is evaluated through a Monte Carlo simulation study comprising B = 500 independent replications with n = 2000 observations per replication, under two engineering-motivated scenarios: a two-component heterogeneous mixture (Weibull–Normal) designed to reflect early-failure and steady-state behavior, and a three-component mixture (Weibull–Normal–Lognormal) representing a full bathtub-curve reliability profile. Model performance is assessed using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The proposed heterogeneous models yield mean AIC improvements of 717.7 and 606.6 units over conventional Gaussian mixture benchmarks for the two- and three-component scenarios respectively, with the heterogeneous model achieving superior fit in all 500 replications in both scenarios. These results provide strong empirical evidence that cross-family heterogeneity in mixture components offers substantial and consistent gains in model fit for engineering data with structurally complex generating processes. Future research should extend the framework to censored lifetime data, formalize component family selection procedures, and investigate Bayesian estimation alternatives.