A New Three-Parameter Weibull Inverse Rayleigh Distribution: Theoretical Development and Applications

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.