Posterior computations based on sample quantiles: one- and two-parameter exponential cases

Abstract

Statistical analyses based on incomplete ordered samples are common in many practical fields. Some data are unknown or disregarded due to restrictions on data collection, experimental difficulties or negligence. In the interest of compressing the data, only several informative order statistics are considered. Subsets of the available data, i.e. training samples, are widely used in a variety of statistical methodologies. In this paper, on the basis of some sample quantiles, a Bayesian analysis of the one- and two-parameter exponential models is developed. Using natural non-informative and conjugate priors, the posterior density and distribution functions for the location and scale parameters are obtained in closed-forms. Explicit expressions for the posterior moments of the exponential parameters and distribution function at any fixed point are derived. The existence and uniqueness of the posterior modes are established; simple and precise lower and upper bounds are also provided. Finally, an illustrative example concerning failure time data is included.