Abstract
In this article we give theoretical results for different stochastic orders of a log-scale-location family which uses Tsallis statistics functions. These results describe the inequalities of moments or Gini index according to parameters. We also compute the mean in the case of q-Weibull and q-Gaussian distributions. The paper is aimed at analyzing the order between survival functions, Lorenz curves and (as consequences) the moments together with the Gini index (respectively a generalized Gini index). A real data application is presented in the last section. This application uses only the survival function because the stochastic order implies the order of moments. Given some supplementary conditions, we prove that the stochastic order implies the Lorenz order in the log-scale-location model and this implies the order between Gini coefficients. The application uses the estimated parameters of a Pareto distribution computed from a real data set in a log-scale-location model, by specifying the Kolmogorov–Smirnov p-value. The examples presented in this application highlight the stochastic order between four models in several cases using survival functions. As direct consequences, we highlight the inequalities between the moments and the generalized Gini coefficients by using the stochastic order and the Lorenz order.
Highlights
The main purpose of this article is to introduce a family of lifetime distributions, called the Tsallis log-scale-location family
We prove that the stochastic order implies the Lorenz order in the log-scale-location model and this implies the order between Gini coefficients
In this article we propose a new generalized log-scale-location family of distributions and we gave results on different stochastic orders for this generalized log-scale-location family that uses the Tsallis statistics
Summary
The main purpose of this article is to introduce a family of lifetime distributions, called the Tsallis log-scale-location family. It has been shown to be very flexible in modeling various types of lifetime data with monotone failure rates but it is not useful for modeling the bathtub shaped and the Mathematics 2021, 9, 1216 unimodal failure rates, which are common in reliability and biological studies It is of utmost interest because of its great number of special features and its ability to fit data from various fields, ranging from life data to observations made in economics and business administration, meteorology, hydrology, quality control, acceptance sampling, statistical process control, inventory control, physics, chemistry, geology, geography, astronomy, medicine, psychology, material science, engineering, biology. Aijaz et al [58] introduced a new Hamza two parameter distribution and studied its properties, including the moments, stochastic orderings, Bonferroni and Lorenz curves, Rényi entropy, order statistics, hazard rate function and mean residual function. A real data application is presented in Section 9 and some general conclusions of the article are given in the last section
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