On the (Hoang) Pham Distribution

The Pham distribution [H. Pham, A vtub-shaped hazard rate function with applications to system safety, Int. J. Reliab. Appl. 3 (2002) 1–16] has not been studied much in spite of being introduced over 20 years ago. Several of it basic mathematical properties are not known in closed form. In this paper, explicit expressions are derived for its mathematical properties, including hazard rate function, reversed hazard rate function, shapes of the probability density, hazard rate and reversed hazard rate functions, stochastic orderings, moment properties, quantiles, mean deviations, mean residual life, expected inactive time, Lorenz, Bonferroni and Zenga curves, order statistics, extreme value statistics, geometric mean, Shannon entropy, Rényi entropy, Tsallis entropy, cumulative residual entropy, maximum likelihood estimation, observed information matrix and expected information matrix. A simulation study is performed to assess finite sample performance of the maximum likelihood estimators. A data application of the Pham distribution is illustrated.