A probablistic analysis for an optimal screening problem

Summary

We consider a lotL formed byN apparently similar unitsW ₁,…,W _N, where each of theW _i may come from one of two different populationsP ₁ andP ₂;T ₁,…,T _N denote the corresponding lifetimes. The units fromP _i undergo a failure of kindi and their survival function isS _i (t).

We assume that the failure rate function$\lambda _i (t) = - \frac{d}{{dt}}\log S_i (t) (i = 1,2)$ are known and that the units fromP ₁ are «substandard»: λ ₁ (t)≥λ ₂ (t), ∀t≥0.

We want to putW ₁,…,W _N under a pre-operational test (burn-in test) in order to eliminate at least a great part of the substandard units and we face the problem of obtaining a rule for stopping the test under the assumption that, with the failure of a unit, it is possible to recognize the population from which the unit comes.

Such a problem will be formalized as an optimal stopping problem for a suitably defined Markov process. Our study shall evidentiate some fundamental aspects of the problem and the role of the prior distribution of the (random) numberM ₀ of those units inL coming fromP ₁ (substandard). The latter distribution has a great influence on the form of the solution.