A new logarithmic Lomax–Rayleigh distribution: properties, simulation, and applications

This paper introduces the logarithmic Lomax–Rayleigh (Log-LR) distribution, a new three-parameter lifetime model within the Lomax–G family, proposed to provide enhanced flexibility for modeling skewed and heavy-tailed data arising in reliability and survival analysis. The model adopts the Rayleigh distribution as the baseline and incorporates two additional shape parameters. The resulting hazard rate function (HRF) is highly flexible and can exhibit inverse-J and non-monotonic decreasing–increasing shapes depending on the parameter values, making the model suitable for a wide range of lifetime data. Several fundamental statistical properties are derived, including the probability density function, cumulative distribution function (CDF), reliability and hazard functions, cumulative and reverse hazard rates, the quantile function, Shannon entropy, and the mean residual life function (MRL). The $r$-th moments and L-moments are expressed in integral form and evaluated numerically. Additional characteristics, including the moment generating function (MGF), skewness, kurtosis, and order statistics, are also investigated. The model parameters are estimated via maximum likelihood, and a Monte Carlo simulation study is conducted to assess the consistency and efficiency of the estimators across different sample sizes. The practical usefulness of the Log-LR distribution is illustrated using four real datasets. Goodness-of-fit comparisons, based on log-likelihood, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Kolmogorov–Smirnov (KS), Anderson-Darling (AD), and Cramér-von Mises (CvM) statistics together with their corresponding $p$-values, are performed against nine competing models. The results consistently demonstrate that the proposed Log-LR distribution outperforms the competing models, highlighting its effectiveness as a flexible alternative for modeling lifetime data.