By Chris P. Tsokos
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BAYESIAN LIFE TESTING USING THE TOTAL Q ON TEST by WILLIAM S. JEWELL kb^tAact. Suppose the basic shape of the cumulative failure (hazard) function has been identified for a certain component, and that an unknown parameter θ for a new production run of similar components is to be estimated. In particular, suppose that the failure function is of ptiopotuLLovwUL type, R(x) = 9Q(x), where Q is the known shape function, and that from a prior gamma density. to tat Q. on toJ>t Θ is sampled By using a new statistic, called the (TQT), it is possible to perform Bayesian up- dating during a variety of lifetime testing programs in a manner similar to total time on test plots.
One seeks to determine a value for approximating the density w, (t) without first g(6). Robbins [86], [87] was able to provide a set of consistent estimators which, when the loss function was bounded in the parameter, were also asymptotically optimal. Samuel [91] developed a set of optimal empirical Bayes rules, based on the estimators considered by Robbins, by which a simple hypothesis could be tested. Using a limit process, he extended the applicability of these rules to the testing of one- and twosided composite hypotheses about the parameter of a discrete failure model.
ReAexVich Log, QaaAtoAly, Vol. 17, (1970), pp. 41-54. [68] and F. E. Grubbs, Approximately Optdirtum Confidence Bounds on Series System Reliability for Exponential Time to Failure Data, BiomQ&Uka, Vol. 59, No. 1, (1972), pp. 191-204. [69] , R. E. Schäfer and N. D. Singpurwalla, Methods for Statistical Analysis of Reliability and Life Data, John Wiley and Sons, New York, (1974). [70] Maritz, J. , Smooth Empirical Bayes Estimation for OneParameter Discrete Distributions, B