Most of the existing life test plans are concerned with life tests conducted under continuous inspection of test items. However, a reduction in testing effort and administrative convenience may be achieved by employing intermittent inspection in which items are inspected only at certain points in time. The well-known hybrid plan, which employs a single inspection at the censoring time, is frequently used in practice. To investigate whether or not adding more inspections to the hybrid plan improves its statistical performance substantially, a life test plan with multiple inspections is first developed assuming that the lifetimes of items follow an exponential distribution. For multiple inspection times, equally-spaced, equal-probability, and equally-spaced in log-time inspection schemes are considered. The developed plan is then compared to the hybrid and the Type I-censored life test plan with continuous inspection in terms of the sample size required, the expected completion time to reach a decision, and the power of test. Computational results indicate that the above performance measures are relatively insensitive to the inspection scheme employed, which implies that the equally-spaced inspection scheme may be considered as an appropriate choice since it is more convenient to schedule and does not require a priori information on unknown parameters. Another important finding is that the life test plan with multiple inspections is preferred when the discrimination ratio is small and the censoring time is large. Otherwise, conducting more than one inspection does not improve statistical performances substantially, and the hybrid plan is generally recommended.
Keywords: Censored Data, Grouped Data, Inspection, Life Testing, Maximum Likelihood Estimator
by Sun-Ho Kim, Agency for Defense Development, Taejon, 305-600 Korea and Bong-Jin Yum, Korea Advanced Institute of Science and Technology, Taejon, 305-701 Korea
A life test plan (LTP) consists of a set of life test procedures and rules for either accepting or rejecting a collection of items based upon the sampled lifetime data. Since an LTP inevitably involves a life test which is time-consuming, it usually employs one or more schemes for managing the testing time at an appropriate level. Among such schemes, censoring is most frequently used. For instance, Epstein and Sobel (1953) developed an LTP to meet the producer and the consumer risks under the assumptions of exponential lifetimes and Type II censoring. Epstein (1954) also proposed the so called hybrid LTP in which Type I and II censoring schemes are combined. Recently, Jeong and Yum (1995) extended the work of Spurrier and Wei (1980) and developed a Type I-censored LTP considering both consumer and producer risks. For other types of LTP, the reader is referred to Lawless (1982), Grant and Leavenworth (1996), and Juran (1988) among others.
Most of the existing LTPs are concerned with life tests conducted under continuous inspection of test items (e.g., Epstein and Sobel (1953), Jeong and Yum (1995)). Although continuous inspection provides more information on the lifetimes of test items, a reduction in testing effort and administrative convenience may be achieved by employing intermittent inspection in which items are inspected only at certain points in time. Further, in some cases, intermittent inspection is the only feasible way of inspecting items due to the limited capability of test equipment (e.g., lack of a device for continuously monitoring the status of test items). For more discussions on the motivation for employing intermittent inspection and/or data examples, the reader is referred to Nelson (1982), Meeker (1986), and Shapiro and Gulati (1996) among others. The resulting data from intermittent inspection are the number of failed items in each of the inspection intervals and are called grouped data in the literature (Kulldorff (1961)). As a special case of intermittent inspection, items may be inspected only once at the censoring time. The hybrid LTP proposed by Epstein (1954) employs such an inspection scheme.
The present investigation was motivated by the authors experiences with industrial practice. As a motivational example, consider an electronics company which manufactures a certain type of voltage regulators. The company requires that all produced items go through a burn-in process so that weak ones that might cause early failures in the field may be eliminated before shipment. The burn-in time was determined such that the failure rate of the items after burn-in becomes reasonably constant (i.e., the lifetimes of the items after burn-in may be considered as exponential). The company is employing an LTP under which the probability of accepting lots with a mean lifetime of 60,000 hours or more is at least 0.95 and the probability of accepting lots with a mean lifetime of 35,000 hours or less is at most 0.1. The life test is conducted for 1,000 hours at a temperature-accelerated condition for which the acceleration factor is known to be 25. This is equivalent to the test duration of 25,000 hours at the use condition. Currently, the company does not have a device which is connected to every item in the chamber to continuously monitor and record its status, and therefore, the hybrid LTP proposed by Epstein (1954) is being used. For the above test situation, the hybrid LTP requires the sample size of 75. The company often asked if adding a few more inspections to the hybrid scheme might improve its statistical performance substantially. To answer this question in an appropriate manner, it is first necessary to develop an LTP with an arbitrary number of inspections and then investigate its performance as a function of the number of inspections.
For the case of exponential lifetimes, Shapiro and Gulati (1996) proposed an LTP in which two inspections are conducted with equiprobable inspection times. The test statistic is based on Y = X3 - X1, where X1 is the number of failures in the first inspection interval, and X3 is the number of items that have not failed by the second inspection time. The sample size and rejection criteria are determined such that the producer and the consumer risks are satisfied. Shapiro and Gulati compared the proposed inspection scheme with others (including the case of continuous inspection) in terms of the sample size required and recommended the former as an adequate choice. Note, however, that the censoring time of the above LTP cannot be controlled by the user and should be always set to , where is the mean lifetime specified in the null hypothesis. In addition, due to the nature of Y, the life test must continue until the last inspection and the possibility of early completion (i.e., curtailment) cannot be incorporated into the plan.
It is also worth noting that Nelson (1977) determined optimal inspection times for the case of exponential lifetimes and multiple inspections such that the large-sample variance of the maximum likelihood estimator of the mean lifetime is minimized. Nelson also developed the locally most powerful test procedure for the purpose of "reliability demonstration." Since the inspection times, including the last one, also depend on the mean lifetime in the above study, the censoring time cannot be controlled by the user.
In this article, the lifetimes of items are assumed to follow an exponential distribution and the maximum likelihood estimation method is employed for estimating the mean lifetime based on the grouped data. The censoring time is assumed to be pre-specified independently of any model parameters (e.g., the mean lifetime). Then, for a given set of parameter values, the sample size and the critical region for the maximum likelihood estimator (MLE) of the mean lifetime are determined such that the consumer and the producer risks are satisfied. Procedures for curtailing a life test are also developed. The resulting LTP is then compared to the hybrid LTP and the Type I-censored LTP with continuous inspection in terms of the sample size required, the expected time to reach a decision, and the power of test.
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