Default Bayesian testing for the scale parameters in two parameter exponential distributions

Abstract

Abstract In this paper, we consider the problem of testing the equality of the scale parametersin two parameter exponential distributions. We propose Bayesian testing proceduresfor the equality of the scale parameters under the noninformative priors. The nonin-formative prior is usually improper which yields a calibration problem that makes theBayes factor to be de ned up to a multiplicative constant. Thus, we propose the defaultBayesian testing procedures based on the fractional Bayes factor and the intrinsic Bayesfactors under the reference priors. Simulation study and an example are provided.Keywords: Fractional Bayes factor, intrinsic Bayes factor, reference prior, scale param-eter, two parameter exponential distribution. 1. Introduction The exponential distribution plays an important role in the eld of life testing and reliabil-ity. One is referred to Zelen (1966), Johnson and Kotz (1970), Bain (1978) and Lawless andSinghal (1980). The probability density function of two parameter exponential distributionE(;˙) with the location parameter and the scale parameter ˙is given byf(xj;˙) =1˙expˆx ˙˙;x>0;˙>0: (1.1)The decision theoretic estimation of the scale parameter was rstly studied by Arnold(1970). Zidek (1973), Brewster (1974), Kubokawa (1994) and Petropoulos and Kourouklis(2002) considered Bayesian estimation of scale parameter based on decision theory. Also,the estimator of the ratio of the scale parameters from the decision theoretic point of viewwas studied by Madi and Tsui (1990), Madi (2008) and Bobotas and Kourouklis (2011). Allthe papers mentioned above were focused on Bayesian point estimation.