Exponentiated Power Ailamujia Distribution: Properties and Applications to Time Series

Abstract

In this study, we propose a new three-parameter distribution, the Exponentiated Power Ailamujia Distribution (EPAD), as a flexible generalization of the Power Ailamujia Distribution (PAD). We derive key statistical properties of the EPAD, including its probability density function (PDF), cumulative distribution function (CDF), survival and hazard rate functions, raw and incomplete moments, quantile function, entropy, stochastic ordering, and order statistics. The PDF is shown to exhibit various shapes—unimodal, J-shaped, and bathtub-shaped—depending on the parameter configurations. A Monte Carlo simulation based on 1,000 replications and seven different sample sizes (n = 25 to 1000) confirms the consistency of the maximum likelihood estimators (MLEs), as root mean square errors (RMSEs) decrease from 0.2131 at n = 25 to 0.1031 at n = 1000. Two real-world datasets, including maximum flood levels and tax revenue data, are modeled using the EPAD and compared against PAD, the Exponentiated Power Lindley Distribution (EPLD), and the Exponentiated Weibull Distribution (EWD). Based on model selection criteria—Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Kolmogorov-Smirnov (KS), Cramer-von Mises (CVM), and Anderson-Darling (AD)—EPAD yields superior fits, with the lowest AIC values of -26.52 and 382.81 for the respective datasets and p-values exceeding 0.90. These results demonstrate the potential of the EPAD in modeling time series data with complex distributional behavior.