The Annals of Statistics

Nonparametric estimation for a general repair model

Crisanto Dorado, Myles Hollander, and Jayaram Sethuraman

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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. Kijima 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. In this paper, 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 for the distribution of the time to first failure of the system.

Article information

Ann. Statist., Volume 25, Number 3 (1997), 1140-1160.

First available in Project Euclid: 20 November 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62N05: Reliability and life testing [See also 90B25]
Secondary: 62G05: Estimation 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx]

Better-than-minimal repair confidence band counting process Markov renewal process product integral supplemented-life-repair weak convergence


Dorado, Crisanto; Hollander, Myles; Sethuraman, Jayaram. Nonparametric estimation for a general repair model. Ann. Statist. 25 (1997), no. 3, 1140--1160. doi:10.1214/aos/1069362741.

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