Sequential Test for Arbitrary Ratio of Mean Times Between Failures

Introduction

The paper deals with development of a planning methodology and tools for truncated comparison SPRT testing of two items.

Such a comparison is effected, for example, in compliance (acceptance) testing, with a view to establishing that:

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  A new item is superior/not inferior in reliability to a predecessor.

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  A new production/repair technology for an item enhances/does not impair its reliability.

In simultaneous testing of the compared items the impact of changing ambient factors is eliminated or at least weakened, which is of special importance in accelerated tests.

Such comparison tests are typical not only in the field of reliability but also in many others, such as maintainability (Moustafa, 2008), efficacy assessment in decision making (Lagroue, 2008; Yeh & Yan, 2010), management (Xiao, Cai, & Jin, 2010).

The items in question are assumed to have exponentially distributed times to failure or times between failures (TBF), whose means are identically termed MTBF. One of these items is termed “new” (subscript n) and the other “basic” (subscript b), Φ being the ratio of their MTBF's:

The test consists in verifying that Φ is not less than a specified value Φ₀:

The exponential-TBF version is very common, as this distribution is the most characteristic in electronics and in complex devices (Kapur & Lamberson, 1977; Kececioglu, 1993; MIL-HDBK-781A, 1996; IEC-61124, 2006; HDBK-217-Plus, 2006). It is also practiced where groups of items are involved, thereby shortening the test duration (Epstein & Sobel, 1955).

Mace (1974) describes planning of such a test with fixed sample sizes (FSST). This version has the disadvantage of predetermined numbers of failures for both items – in which case if one item turns out greatly superior to the other, it will take very long for it to accumulate the necessary volume – whereas common sense would indicate an end to the test much sooner. Another disadvantage is the much larger sample sizes (or sample numbers – SN, see Wald (1947)) than the average requirement of the sequential probability ratio test (SPRT).

Girshick (1946) applies the SPRT theory in comparison tests of two items, the disadvantage here being the need for pairwise observation. A planning method for determining the scale-parameter ratio of two exponential distributions is proposed by Uno (2003). This problem differs from the one presented in this paper in that in our case it is not required to establish the actual value, but only that it is not less than a prescribed one. Uno's solution calls for not less than two failures per item, which substantially impairs its efficacy when the scale parameters are widely different. The solution is approximative and has a significant bias. Zacks and Mukhopadhyay (2007) deal with an analogous problem of boundary construction and deciding on the MTBF ratio of two systems with exponential lifetimes. Their solution is exact, but their problem likewise differs from ours – namely, a ratio not less than a prescribed value.

In Michlin and Grabarnik (2007) the test under discussion is reduced to a binomial SPRT – one of the oldest described by Wald (1947). In Wald and Wolfowitz (1948) proof is provided of the optimality of the SPRT at two values of the estimated parameter – in this case Φ₀ and Φ₁,which in turn correspond to the null (H₀) and alternative (H₁) hypotheses respectively – and for which the operating characteristic (OC) has the values (1–α) and β. α and β being the respective probabilities of type I and II errors. As for the optimality criterion, it is a minimal average SN (ASN) to stopping the test at given α and β.