Bayesian Estimation of the Scale Parameter of Inverse Weibull Distribution under the Asymmetric Loss Functions

This paper proposes different methods of estimating the scale parameter in the inverse Weibull distribution (IWD). Specifically, the maximum likelihood estimator of the scale parameter in IWD is introduced. We then derived the Bayes estimators for the scale parameter in IWD by considering quasi, gamma, and uniform priors distributions under the square error, entropy, and precautionary loss functions. Finally, the different proposed estimators have been compared by the extensive simulation studies in corresponding the mean square errors and the evolution of risk functions. 1. Introduction It is well known that the Weibull distribution is one of the most popular distributions in the lifetime data analyzing. The main reason is that one can create a wide variety of shapes with varying levels of its parameters. Therefore, during the past decades, extensive work has been done on this distribution in both the frequentist and Bayesian points of view; see, for example, the excellent reviews by Johnson et al. [1] and Kundu [2]. However, the Weibull distribution has two parameters, and in many practical applications, one or both of them might be unknown. To estimate them, we may use common approaches (see, e.g., Nordman and Meeker [3]). Moreover, it is clear through the distribution of Weibull that the Weibull probability density function (PDF) can be decreasing (or increasing) or unimodal, depending on the shape of distribution parameters. Due to the flexibility of the Weibull PDF, the inverse Weibull distribution (IWD) has been extensively employed in situation where a monotone data set is available (REF). Furthermore, if the empirical studies indicate that the Weibull PDF might be unimodal, then the inverse Weibull distribution (IWD) may be an appropriate model (Kundu [2]). As a definition, if a positive random variable has the Weibull distribution with the following PDF: then the random variable has the IWD with the PDF of the following form: where is called scale parameter and is called shape parameter of this family. It also follows from (2) that the cumulative distribution function of can be obtained: IWD plays an important role in many applications, including the dynamic components of diesel engines and several data sets such as the times to breakdown of an insulating fluid subject to the action of a constant tension (see Drapella [4], Jiang et al. [5], and Nelson [6] for more practical applications). For instance, Calabria and Pulcini [7] provide an interpretation of the IWD in the context of the load-strength relationship for a component. Maswadah [8]