Abstract

In this work, a three-parameter Weibull Inverse Rayleigh (WIR) distribution is proposed. The new WIR distribution is an extension of a one-parameter Inverse Rayleigh distribution that incorporated a transformation of the Weibull distribution and Log-logistic as quantile function. The statistical properties such as quantile function, order statistic, monotone likelihood ratio property, hazard, reverse hazard functions, moments, skewness, kurtosis, and linear representation of the new proposed distribution were studied theoretically. The maximum likelihood estimators cannot be derived in an explicit form. So we employed the iterative procedure called Newton Raphson method to obtain the maximum likelihood estimators. The Bayes estimators for the scale and shape parameters for the WIR distribution under squared error, Linex, and Entropy loss functions are provided. The Bayes estimators cannot be obtained explicitly. Hence we adopted a numerical approximation method known as Lindley's approximation in other to obtain the Bayes estimators. Simulation procedures were adopted to see the effectiveness of different estimators. The applications of the new WIR distribution were demonstrated on three real-life data sets. Further results showed that the new WIR distribution performed credibly well when compared with five of the related existing skewed distributions. It was observed that the Bayesian estimates derived performs better than the classical method.

Highlights

  • In probability theory and statistics, a generalized lifetime model suitable for fitting survival and engineering data is often of interest in the survival and reliability analysis [1]

  • Let T follows the Weibull distribution with Cumulative Distribution Function (CDF) given by FT(t) = 1 − exp(−αtβ) its corresponding probability density function (PDF) is given by fT(t) = αβtβ−1exp(−αtβ); α, β > 0; t ≥ 0

  • Let R be a random variable that follows the Inverse Rayleigh distribution with CDF given by GR(r) = exp(−(δ/r)2).‍‍‍‍

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Summary

A New Three-Parameter Weibull Inverse Rayleigh Distribution

Adeyinka Solomon Ogunsanya1,*, Waheed Babatunde Yahya, Taiwo Mobolaji Adegoke, Christiana Iluno, Oluwaseun R. Cite This Paper in the following Citation Styles (a): [1] Adeyinka Solomon Ogunsanya, Waheed Babatunde Yahya, Taiwo Mobolaji Adegoke, Christiana Iluno, Oluwaseun R. Matthew Iwada Ekum , "A New Three-Parameter Weibull Inverse Rayleigh Distribution: Theoretical Development and Applications," Mathematics and Statistics, Vol 9, No 3, pp. (b): Adeyinka Solomon Ogunsanya, Waheed Babatunde Yahya, Taiwo Mobolaji Adegoke, Christiana Iluno, Oluwaseun R. A New Three-Parameter Weibull Inverse Rayleigh Distribution: Theoretical Development and Applications. Mathematics and Statistics, 9(3), 249 - 272.

Introduction
Theoretical Development
Properties of the New WIR Distribution
The Survival Function
The Hazard Function
The Reverse Hazard Function
The Skewness and Kurtosis
Linear Representation
The Monotone Likelihood Ratio Property of WIR Distribution
The Moment
The Maximum Likelihood Estimation
The Quantile Function Estimation Method
The Bayesian Estimation
The Prior Distribution
The Lindley’s Approximation
Simulation Study
Applications
Full Text
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