Some common and dynamic properties of logarithmic Pareto distribution with applications

The Pareto distribution satisfies the power law, which is widely used in physics, biology, earth and planetary sciences, economics, finance, computer science, and many other fields. In this article, the logarithmic Pareto distribution, a logarithmic transformation of the Pareto distribution, is presented and studied. The moments, percentiles, skewness, kurtosis, and some dynamic measures such as hazard rate, mean residual life, and quantile residual life are discussed. The parameters were estimated by quantile and maximum likelihood methods. A simulation study was conducted to investigate the efficiency, consistency, and behavior of the maximum likelihood estimator. Finally, the proposed distribution was fitted to some datasets to show its usefulness.

Some new results on bathtub-shaped hazard rate models

<abstract><p>The most common non-monotonic hazard rate situations in life sciences and engineering involves bathtub shapes. This paper focuses on the quantile residual life function in the class of lifetime distributions that have bathtub-shaped hazard rate functions. For this class of distributions, the shape of the $ \alpha $-quantile residual lifetime function was studied. Then, the change points of the $ \alpha $-quantile residual life function of a general weighted hazard rate model were compared with the corresponding change points of the basic model in terms of their location. As a special weighted model, the order statistics were considered and the change points related to the order statistics were compared with the change points of the baseline distribution. Moreover, some comparisons of the change points of two different order statistics were presented.</p></abstract>

Dynamic Multivariate Quantile Residual Life in Reliability Theory

We extend the univariate α-quantile residual life function to multivariate setting preserving its dynamic feature. Principal attributes of this function are derived and their relationship to the dynamic multivariate hazard rate function is discussed. A corresponding ordering, namely, α-quantile residual life order, for random vectors of lifetimes is introduced and studied. Based on the proposed ordering, a notion of positive dependency is presented. Finally, a discussion about conditions characterizing the class of decreasing multivariate α-quantile residual life functions is pointed out.