Arctan–Weibull Log Logistic Mixture Distribution: Theory and Applications to Lifetime Data

Abstract

This paper introduces the ArcTangent Weibull Log Logistic Mixture (AT-WLLM) distribution, a novel and flexible lifetime model constructed by applying an arctangent transformation to a mixture of Weibull and log-logistic distribution functions. The proposed model exhibits several desirable properties, including the ability to capture a wide range of hazard rate shapes commonly encountered in reliability engineering and survival analysis. The theoretical properties of the AT-WLLM distribution are thoroughly investigated. Parameter estimation is carried out using the maximum likelihood method, and a comprehensive simulation study is conducted to assess the performance of the estimators. Across all parameter configurations considered, the MLE procedure for the AT-WLLM distribution exhibits numerical stability and satisfactory finite–sample performance. The practical applicability of the model is demonstrated through three real-data analyses, in which the AT-WLLM distribution is compared with seven well-known competing models. The empirical results consistently affirm the superior flexibility and strength of the AT-WLLM distribution in modeling lifetime data.