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The properties of nonparametric estimations of probability characteristics of system nonfailure operation with taking into account repairable failures. (English. Russian original) Zbl 0838.62092

Mosc. Univ. Math. Bull. 48, No. 5, 25-29 (1993); translation from Vestn. Mosk. Univ., Ser. I 1993, No. 5, 27-32 (1993).
Summary: Regenerating processes of Poisson type are considered, in which to each cycle, up to the next moment of regeneration, there corresponds a Poisson point process with a continuous conductor function \(H(s)\), restricted at any value \(s\). The moment of regeneration coincides with the moment \(t_k\) of occurrence of the \(k\) th event in the cycle. The observation data are constituted by independent realizations of such regenerating processes, considered on the interval \([0,t ]\). Nonparametric estimations of \(H(s)\), \(s\in [0,t ]\), are obtained, their consistency and \(\sqrt {m}\)-asymptotic normality are proved. A formulation of the problem in terms of the reliability theory is given. The presented example illustrates the effectiveness of the proposed estimations.

MSC:

62N05 Reliability and life testing
62G05 Nonparametric estimation
62M09 Non-Markovian processes: estimation
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