Ann. Inst. Statist. Math. Vol. 44, No. 4, 623-639 (1992) BAYESIAN INFERENCE FOR THE POWER LAW PROCESS SHAUL K. BAR-LEV, IDIT LAVl AND BENJAMIN REISER Department of Statistics, University of Haifa, Haifa 31999, Israel (Received December 17, 1990; revised November 8, 1991) Abstract. The power law process has been used to model reliability growth, software reliability and the failure times of repairable systems. This article re- views and further develops Bayesian inference for such a process. The Bayesian approach provides a unified methodology for dealing with both time and failure truncated data, As well as looking at the posterior densities of the parameters of the power law process, inference for the expected number of failures and the probability of no failures in some given time interval is discussed. Aspects of the prediction problem are examined. The results are illustrated with two data examples. Key words and phrases: Power law process, Bayesian inference, prediction, repairable system. 1. Introduction The power law process can be described as a nonhomogeneous Poisson process {N(t), t >_ 0} with intensity function u(t) =/3t~-1/~ ~, c~ > 0, ~q > 0, and mean value function re(t) = E(N(t)) = (t/a) ~. This has been widely used in the litera- ture to model reliability growth (Crow (1982)), software reliability (Kyparisis and Singpurwalla (1985)), and more generally repairable systems (Ascher and Feingold (1984), Engelhardt and Bain (1986), Rigdon and Basu (1989)). We follow Ascher (1981) in using the term power law process being convinced by his arguments that this is superior to the more frequently used Weibull process terminology. An alternate way of describing the power law process is to consider the se- quence of successive failure times T1, T2,..., where Ti is the time of the i-th fail- ure. Then the time of the first failure T1 has a Weibull distribution with scale and shape parameters c~ and/3, respectively, while the failure time Ti, conditional on T1 = Q,..., Ti-1 = ti-1, has a truncated Weibull distribution with left truncation point ti-1. Inference on the power law process has generally been considered in the liter- ature from a frequency viewpoint. Two sampling schemes are usually considered; failure truncation and time truncation. For these sampling schemes the large lit- erature on point estimation, confidence intervals, and tests of hypotheses for the parameters c~, /3, and the intensity function at the end of the testing period is 623