# Closed-form mathematical expressions for the exponentiated Cauchy-Rayleigh distribution

## TL;DR

This paper introduces the exponentiated Cauchy-Rayleigh (ECR) distribution, a flexible lifetime model with closed-form expressions and multiple hazard rate shapes, outperforming classical models in heavy-tail data analysis.

## Contribution

The paper proposes the ECR distribution with new closed-form expressions and estimation methods, enhancing modeling of asymmetric, heavy-tail lifetime data.

## Key findings

- ECR distribution has versatile hazard rate shapes.
- Simulation shows estimators perform well.
- Application demonstrates ECR's superiority over classical models.

## Abstract

The Cauchy-Rayleigh (CR) distribution has been successfully used to describe asymmetric and heavy-tail events from radar imagery. Employing such model to describe lifetime data may then seem attractive, but some drawbacks arise: its probability density function does not cover non-modal behavior as well as the CR hazard rate function (hrf) assumes only one form. To outperform this difficulty, we introduce an extended CR model, called exponentiated Cauchy-Rayleigh (ECR) distribution. This model has two parameters and hrf with decreasing, decreasing-increasing-decreasing and upside-down bathtub forms. In this paper, several closed-form mathematical expressions for the ECR model are proposed: median, mode, probability weighted, log-, incomplete and order statistic moments and Fisher information matrix. We propose three estimation procedures for the ECR parameters: maximum likelihood (ML), bias corrected ML and percentile-based methods. A simulation study is done to assess the performance of estimators. An application to survival time of heart problem patients illustrates the usefulness of the ECR model. Results point out that the ECR distribution may outperform classical lifetime models, such as the gamma, Birnbaun-Saunders, Weibull and log-normal laws, before heavy-tail data.

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Source: https://tomesphere.com/paper/1706.03862