Quasi-Parametric Bayes Estimates of the TTF Probability

Abstract

A quasiparametric Bayes estimate plays the intermediate role between the parametric and nonparametric estimates in the following sense. Let {Ω,ι,F} be a sample space, characterizing a statistical model, generated by some plan P; a probability measure F (distribution) which belongs to some class S. A quasiparametric estimate of a time to failure probability r, connected functionally with F, is constructed in the following manner. Starting with given a priori information, the probability measure is approximated with the help of 3F Є Sθ , where Sθ is a parametric family chosen in accordance with the form of representation of a priori information, where Sθ Є S. The parameter q is random and the space{Θ,ε ,H} where H Є H is a prior probability measure of the parameter θ on (Θ,ε ). The estimate of the probability of a TTF is sought in the form of the Bayes estimate of the corresponding function R=R(θ ), measurable on θ . A specific Bayes procedure is determined by the form of a priori information and by the chosen approximation of c.d.f., F, on the class S. These estimates are proposed and investigated in the works. Consider some important aspects from a practical point-of-view, cases dealing with a construction of the approximation of the unknown c.d.f., F(t), on the classes of failure rate Sθ or failure rate in mean S ₁ distributions. Throughout what follows we will use a representation of the approximation distribution function with the help of the resource function: