Alpha-Power Exponentiated Inverse Rayleigh distribution and its applications to real and simulated data.

Feng Chen,Alamgir Khalil,Muhammad Ali,Muhammad Ijaz,Noor Saeed

doi:10.1371/journal.pone.0245253

Abstract

The main goal of the current paper is to contribute to the existing literature of probability distributions. In this paper, a new probability distribution is generated by using the Alpha Power Family of distributions with the aim to model the data with non-monotonic failure rates and provides a better fit. The proposed distribution is called Alpha Power Exponentiated Inverse Rayleigh or in short APEIR distribution. Various statistical properties have been investigated including they are the order statistics, moments, residual life function, mean waiting time, quantiles, entropy, and stress-strength parameter. To estimate the parameters of the proposed distribution, the maximum likelihood method is employed. It has been proved theoretically that the proposed distribution provides a better fit to the data with monotonic as well as non-monotonic hazard rate shapes. Moreover, two real data sets are used to evaluate the significance and flexibility of the proposed distribution as compared to other probability distributions.

Highlights

The development of new distributions has become a common practice in recent decades; this is done generally by adding an extra parameter [1] to the baseline distribution, using generators [2, 3], or by combining two distributions [4]
Lee et al [3] developed a technique of generating single variable continuous distributions
We considered the Exponentiated Inverse Rayleigh distribution as a baseline distribution presented by Rehman and Sajjad [12]

Summary

Introduction

The development of new distributions has become a common practice in recent decades; this is done generally by adding an extra parameter [1] to the baseline distribution, using generators [2, 3], or by combining two distributions [4]. Because the existing distribution has some limitations so model the complex data structures, for example, Exponential and Weibull distributions fail the real data following a non-monotonic failure rate functions. In this aim of presenting the paper is to contribute a new probability distribution that will model the data with both monotonically and non-monotonically hazard rate functions. By applying the cumulative distribution function of the Exponentiated inverse Rayleigh distribution to the ALPF, we obtained the following Cdf and Pdf for the APEIR and is given by FAPEIRðxÞ.

À v 0 a À 1 x3

Conclusion