Interpolational Bayesian estimation of reliability. Binomial trials (Q1342545)

Binomial trials of items for assessing reliability have the following features: the trials are independent in the probability sense; the duaration \(\tau\) of trials is the same for all the items; the results of the trials are given in the form: ``success -- failure''; the parameter subject to estimation is the probability \(\theta\), \(\theta\leq 1\), of an unfailing operation (PUO) of an item during time \(\tau\), i.e., its reliability. The result of binomial trials can be represented by means of a sample \(X_ n= \{x_ 1, x_ 2,\dots, x_ n\}\) when \(n\) is the number of trials, \(x_ i\) is the random variable, \(x_ i=0\) if the \(i\)- th item did not fail, \(x_ i=1\) -- otherwise, \(i= 1,\dots, n\). The problem is to determine a statistic \(\delta= \delta(X_ n)\) which could be used approximately in place of \(\theta\) taking into account the information contained in the sample \(X_ n\). We present -- within the Bayesian framework -- a method for estimating reliability under a binomial sampling plan in the absence of a priori information. Reliability is determined from the aspect of point estimation which is a more difficult statistical problem then the interval estimation.