Nonparametric estimation for a general repair model. (English) Zbl 0937.62103

Dorado, Crisanto; Hollander, Myles; Sethuraman, Jayaram

Ann. Stat. 25, No. 3, 1140-1160 (1997).

Summary: The construction and analysis of repair models is an important area in reliability. A commonly used model is the minimal repair model. Under this model, repair restores the state of the system to its level prior to failure. M. Kijima [J. Appl. Probab. 26, No. 1, 89-102 (1989; Zbl 0671.60080)] introduced repair models that could be classified as “better-than-minimal.” Under Kijima’s models, the system, upon repair, is functionally the same as a working system of lesser age which has never experienced failure.
We present a new approach to the modeling of better-than-minimal repair models. Using this approach, we construct a general repair model that contains Kijima’s models as special cases. We also study the problem of estimating the distribution of the time to first failure of a system maintained by general repair. We make use of counting processes to show strong consistency of the estimator and prove results on weak convergence. Finally, we derive a Hall-Wellner type asymptotic confidence band [W. J. Hall and J. A. Wellner, Biometrika 67, 133-143 (1980; Zbl 0423.62078)] for the distribution of the time to first failure of the system.

MSC:

62N05	Reliability and life testing
62G05	Nonparametric estimation
60K20	Applications of Markov renewal processes (reliability, queueing networks, etc.)

Keywords:

confidence band; Markov renewal processes; product integrals; supplemented-life-repair; better-than-minimal repair; counting processes; weak convergence

Citations:

Zbl 0671.60080; Zbl 0423.62078