Efficient estimation of mean time to failure

Product testing plan of type has been chosen as the subject of research plan. This plan's time between failures is subject to the exponential law where N is the number of same-type tested products; T is the time to failure (same for each product); is a feature of the plan meaning that after each failure the working condition of the product is recovered over the course of the test. In this case, the time to failure is defined according to formula T01 = NT/ , where is the number of observed failures, ω > 0, that occurred within the time T. This estimate is biased. Besides that, if it is required to solve a problem that involves achieving a point estimation of mean time to failure (T0) of products based on tests that did not produce any failures, estimate T01 cannot be used. If over the time of testing the number of observed failures is small (the number does not exceed several ones), the estimate can contain a significant error due to the bias. In order to solve the above problem, it suffices to find an unbiased efficient estimate T0ef of the value T0, if such exists, in the class of consistent biased estimates (the class of consistent estimates that includes all estimates generated by method of substitution, of which the maximum likelihood method, contains estimates with any bias, including those with a fixed one, in the form of function of parameter or constant). In general, there is currently no rule for finding unbiased estimates, and their identification is a sort of art. In some cases, the generated unbiased efficient estimates are quite lengthy and have a complex calculation algorithm. They are also not always sufficiently efficient in the class of all biased estimates and not always have a considerable advantage over simple yet biased estimates from the point of view of proximity to the estimated value.

The aim of the article is to find the estimate of value T0 that is simple and more efficient in comparison with the conventional one and negligibly inferior to the estimate T0ef, if such exists, in terms of proximity to T0 when using the NMT plan.

Methods. In obtaining an efficient estimate integral characteristics were used, i.e. total relative square of the deviation of expected realization of estimate T0ω from various values T0 per various failure flows of the tested product population. A sufficiently wide range of class estimates was considered and a functional built based on the integral characteristic, of which the solution finally allowed deducing a simple and efficient evaluation of mean time to failure for the NMT plan.

Conclusions. The achieved estimate of mean time to failure for the NMT plan is efficient within a sufficiently wide range of estimates and is not improvable within the considered class of estimates. Additionally, the achieved estimate enables point estimation of mean time to failure based on the results of tests that did not have any failures.

1. GOST 27.003–90. Technology dependability. Selection and standardization of dependability indicators. General provisions.

2. Barzilovich, E.Yu., Beliaev, Yu.K., Kashtanov, V.A. et al. Matters of mathematical theory of dependability. edited by Gnedenko, B.V. – Moscow: Radio i sviaz, 1983. – 376 p.

3. Zarenin, Yu.G., Stoyanova, I.I. Determinative tests of dependability. – Moscow: Izdatelstvo standartov, 1978. – 168 p.

4. Levin, B.R. Dependability theory of radiotechnical systems (mathematical foundations). Study guide for colleges. Moscow, Sov. radio, 1978. – 264 p.

5. Borovkov, A.A. Mathematical statistics. – Moscow: Nauka, 1984. – 472 p.

6. Krol, I.А. Unbiased estimates for quantitative dependability characteristics under various test plans//Matters of dependability of integrated electromechanics systems, static converters and electric machines: Proceedings of VNIIEM. — Moscow, 1970. – 231 p.

7. Beliaev, Yu.K, Zamiatin, А.А. On unbiased estimates of the exponential distribution parameter// Journal of the ASUSSR. Engineering cybernetics. – 1982. – Issue No. 3. – P. 92-95.

8. Voinov, V.G., Nikulin, M.S. Unbiased estimates and their application. – Moscow: Nauka, 1989.

9. Lumelsky, Ya.P., Shekhovtsova, M.G. On the comparison of estimated probabilities of fault-free operation under Poisson failure flow// Nadiozhnost i kontrol kachestva. – 1986. – Issue No. 9. – P. 17–22.

10. Shekhovtsova, M.G. Integral characteristics of dependability monitoring plans under Poisson failure flow/ Perm State University. – Perm, 1983. – 13 p. – Reference: 5 titles. – Deposited with VINITI on 15.11.85, No. 7955-V.

11. Mikhailov, V.S. Efficient estimation of mean time to failure // Nadiozhnost i kontrol kachestva. – 1988. – Issue No. 9. – P. 6-11.

12. Gnedenko, B.V., Beliaev, Yu.K., Soloviev, A.D. Mathematical methods in the dependability theory. – Moscow: Nauka, 1965. – 524 p.

13. Matveev, V.F., Ushakov, V.G. Mass service systems. – Moscow: Moscow State University publishing house, 1984. – 239 p.