Type II Topp–Leone Inverted Kumaraswamy Distribution with Statistical Inference and Applications

Ramadan A Zeineldin,Christophe Chesneau,Farrukh Jamal,Mohammed Elgarhy

doi:10.3390/sym11121459

Abstract

In this paper, we present and study a new four-parameter lifetime distribution obtained by the combination of the so-called type II Topp–Leone-G and transmuted-G families and the inverted Kumaraswamy distribution. By construction, the new distribution enjoys nice flexible properties and covers some well-known distributions which have already proven themselves in statistical applications, including some extensions of the Bur XII distribution. We first present the main functions related to the new distribution, with discussions on their shapes. In particular, we show that the related probability density function is left, right skewed, near symmetrical and reverse J shaped, with a notable difference regarding the right tailed, illustrating the flexibility of the distribution. Then, the related model is displayed, with the estimation of the parameters by the maximum likelihood method and the consideration of two practical data sets. We show that the proposed model is the best one in terms of standard model selection criteria, including Akaike information and Bayesian information criteria, and goodness of fit tests against three well-established competitors. Then, for the new model, the theoretical background on the maximum likelihood method is given, with numerical guaranties of the efficiency of the estimates obtained via a simulation study. Finally, the main mathematical properties of the new distribution are discussed, including asymptotic results, quantile function, Bowley skewness and Moors kurtosis, mixture representations for the probability density and cumulative density functions, ordinary moments, incomplete moments, probability weighted moments, stress-strength reliability and order statistics.

Highlights

The detailed explanation of a lifetime phenomenon often follows from a deep data analysis based on a well-chosen statistical model
Let us show some basics of the type II Topp–Leone-G family of distributions introduced by [10]
We introduce the type II Topp–Leone inverted Kumaraswamy (TIITLIK)

Summary

Introduction

The detailed explanation of a lifetime phenomenon often follows from a deep data analysis based on a well-chosen statistical model. Symmetry 2019, 11, 1459 the development of new probability distributions. The mathematical techniques for creating new probability distributions are numerous. A common technique is to introduce one or several additional tuning parameters to a standard probability distribution, with the aim to improve it, in the theoretical and practical sense. We refer the reader to the following families of distributions: the exponentiated-G (or exp-G) family [1], the beta-G family [2], the Marshall-Olkin-G family [3], the gamma-G family [4], the Kumaraswamy-G family [5], the type I half-logistic-G family [6], the transmuted-G family [7], the odd power Cauchy family [8], the exponentiated generalised Topp–Leone-G family [9], the type II. Let us show some basics of the type II Topp–Leone-G family of distributions introduced by [10]. The corresponding cumulative distribution function (cdf) and probability density function (pdf) are, respectively, given by n oα

Objectives

Results

Conclusion