The Weibull–Odd Perks Distribution: Statistical Theory, Regression Modeling, and Applications

We introduce the Weibull–Odd Perks distribution (WOPD), a three-parameter lifetime model obtained by composing the odds function of the Perks distribution with the Weibull-G generator. The construction admits a structural reduction thus if X∼WOPDψ,υ,γ, then eυX−1 follows a two-parameter Weibull law, and this identity drives the entire analytic development. Exploiting it, we obtain the quantile function in a closed form, together with exact series representations of the r-th raw moment expressed through the unsigned Stirling numbers of the first kind and parametric derivatives of the upper incomplete gamma function, and an equivalent single-integral form well suited to Gauss–Laguerre quadrature. The Lorenz and Bonferroni curves are derived as lower-incomplete-gamma series, the Shannon entropy is reduced to a unit-exponential expectation involving the Euler–Mascheroni constant, and the Rényi entropy is obtained in a closed form. We further characterize the shape regimes of the model, showing that the density admits unimodal, right-skewed, left-skewed, and approximately symmetric forms while the hazard rate accommodates increasing, decreasing, bathtub, and unimodal behavior. Seven frequentist estimators, namely, maximum likelihood, Anderson–Darling, Cramér–von Mises, maximum product of spacings, ordinary and weighted least squares, and percentile estimation, are compared through Monte Carlo experiments that establish the consistency and finite-sample accuracy of the estimation procedure. A log-location-scale regression extension is developed and validated via Cox–Snell residual diagnostics. In applications to two real lifetime datasets, the WOPD is preferred over eight competing models on all goodness-of-fit and information criteria for one dataset and matches the fit of a four-parameter alternative with one fewer parameter for the other; a further application to hypertension survival data from Ghana illustrates the regression model in a biomedical setting with covariates.