A Flexible Lifetime Distribution Based on Alpha Power Transformation: Properties, Inference and Data Analysis

The Rayleigh–Logarithmic distribution provides a useful framework for modelling lifetime data by combining continuous lifetime variability with a logarithmic compounding mechanism. This study introduces a three-parameter Alpha Power Rayleigh–Logarithmic (APRL) distribution by applying the Alpha Power transformation to the classical Rayleigh–Logarithmic model. The additional transformation parameter allows the distributional shape, skewness, tail behaviour, and rate of increase in the hazard function to be adjusted while retaining the underlying structure of the baseline model. Several mathematical and reliability properties of the APRL distribution are derived, including the probability density and cumulative distribution functions, survival and hazard rate functions, quantile function, moments, order statistics, and mean residual life function. Model parameters are estimated by maximum likelihood using a multiple-start numerical optimization procedure, and the finite-sample performance of the estimators is investigated through Monte Carlo simulations under different parameter configurations and sample sizes. The simulation results show that estimation accuracy generally improves with increasing sample size, as reflected by decreasing bias, MSE, and RMSE, although estimation of the transformation parameter may exhibit greater variability for more extreme parameter settings. The practical performance of the APRL distribution is examined using the Aircraft Windshield Failure Times and Breaking Stress of Carbon Fibres datasets. Model comparisons based on information criteria, bootstrap-based goodness-of-fit assessment, and graphical diagnostics show that the APRL distribution provides competitive fits relative to several established lifetime distributions. In addition, mean time to failure and mean residual life analyses illustrate the practical interpretation of the reliability measures derived for the proposed model. Overall, the results support the APRL distribution as a useful alternative for the statistical analysis of lifetime and reliability data.