Abstract
A three-parameter Maxwell-Mukherjee Islam distribution was proposed by applying Maxwell generalized family of distributions introduced by Ishaq and Abiodun [17]. The probability density and cumulative distribution functions of the proposed distribution were defined. The validity test was derived from its cumulative distribution function. The study aimed to obtain a Bayesian estimation of the scale parameter of Maxwell-Mukherjee Islam distribution by using assumptions of the Extended Jeffrey's (Uniform, Jeffrey's and Hartigan's), Inverse-Rayleigh and Inverse-Nakagami priors under the loss functions, namely, Squared Error Loss Function (SELF), Precautionary Loss Function (PLF) and Quadratic Loss Function (QLF), and their performances were compared. The posterior distribution under each prior and its corresponding loss functions was derived. The performance of the Bayesian estimation was illustrated from the basis of quantile function by using a simulation study and application to real life data set. For different sample sizes and parameter values, the QLF and SELF under Jeffrey's and Hartigan's priors produced the same estimates, bias and Mean Squared Error (MSE) just as we observed in their mathematical derivatives. Similarly, the SELF, PLF and QLF under Inverse-Rayleigh and Inverse-Nakagami priors provided the same performance when some parameter values are equal. For some parameter values, the QLF under Inverse-Nakagami and Inverse-Rayleigh priors produced the least values of MSE. In the application to real life data set, the QLF and SELF under Jeffrey's and Hartigan's priors; the SELF, PLF and QLF under Inverse-Rayleigh and Inverse-Nakagami priors provided similar results as observed in the simulation study. Therefore, the study concluded that the QLF under Inverse-Rayleigh and Inverse-Nakagami priors could effectively be used in the estimation of scale parameter of Maxwell-Mukherjee Islam distribution using Bayesian approach.
Highlights
Mukherjee and Islam [21] developed a continuous statistical distribution popularly known as Mukherjee-Islam distribution that has a finite range of values suitable for modeling datasets related to failure time, reliability analysis, and life testing among others
A three-parameter lifetime statistical distribution referred to as Maxwell-Mukherjee Islam (MMI) distribution is proposed in this study which serves as an extension of Mukherjee-Islam distribution by applying Maxwell generalized family of distributions developed by Ishaq and Abiodun [17]
This study considers the Extended Jeffrey’s (Uniform, Jeffrey’s and Hartigan’s), Inverse-Rayleigh and Inverse-Nakagami priors as the prior distributions
Summary
Mukherjee and Islam [21] developed a continuous statistical distribution popularly known as Mukherjee-Islam distribution that has a finite range of values suitable for modeling datasets related to failure time, reliability analysis, and life testing among others. A three-parameter lifetime distribution was introduced by Rather [24] by adding an extra shape parameter to study Exponentiated Mukherjee-Islam distribution Several properties of this distribution including moments, harmonic mean, moment generating function, Renyi, and Shanon entropies were discussed. A three-parameter lifetime statistical distribution referred to as Maxwell-Mukherjee Islam (MMI) distribution is proposed in this study which serves as an extension of Mukherjee-Islam distribution by applying Maxwell generalized family of distributions developed by Ishaq and Abiodun [17]. This study is organized as follows: In section 2, we define the cumulative distribution and probability density function of the MaxwellMukherjee Islam distribution by using the Maxwell generalized family of distributions.
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