On the method of pivoting the CDF for exact confidence intervals with illustration for exponential mean under life-test with time constraints

Abstract

Two requirements for pivoting a cumulative distribution function (CDF) in order to construct exact confidence intervals or bounds for a real-valued parameter θ are the monotonicity of this CDF with respect to θ and the existence of solutions of some pertinent equations for θ. The second requirement is not fulfilled by the CDF of the maximum likelihood estimator of the exponential scale parameter when the data come from some life-testing scenarios such as type-I censoring, hybrid type-I censoring, and progressive type-I censoring that are subject to time constraints. However, the method has been used in these cases probably because the non-existence of the solution usually happens only with small probability. Here, we illustrate the problem by giving formal details in the case of type-I censoring and by providing some further examples. We also present a suitable extension of the basic pivoting method which is applicable in situations wherein the considered equations have no solution.

On generalized progressive hybrid censoring in presence of competing risks

“…It should be mentioned that the exact confidence intervals of θ 1 and θ 2 may not always exist, see for example Balakrishnan et al [4] for all 0 < α < 1. In fact it is clear that if P θ 1 ( θ 1 ≤ θ 1,obs ) varies from 1 to 0, as θ 1 varies from 0 to infinity, for all values of θ 1,obs , then the exact confidence interval of θ 1 exists for all values of 0 < α < 1.…”

The step-stress tampered failure rate model under interval monitoring

“…. , s) and no intermediate inspection points, (6.3) coincides with the system of equations (5) in Xiong and Ji [42]. Further on, for the simple SSALT model, the system (6.3) leads tô…”

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Stage life testing

“…Casella & Berger, , pp. 430–435, Balakrishnan, Cramer, & Iliopoulos, , Hahn, Meeker, & Escobar, ), we need the survival functions of and , respectively. These survival functions are given in Corollary for both options Type‐P and Type‐M.…”