Yefim Haim Michlin

Sequential Test for Arbitrary Ratio of Mean Times Between Failures

This paper describes a planning methodology and tools for a truncated SPRT (sequential probability ratio test) for checking the means ratio of the times between failures (assumed to be exponentially distributed) of two items. The problem is considered for the situation in which the ratio may differ from unity, whereby the results are applicable for any specified ratio, or wherever multiple copies of each item are tested simultaneously (group test). The authors present a methodology for optimal test choice and dependences for determining the acceptance/rejection boundaries of such a test with given characteristics. Planning and implementation of a group test are illustrated in an example, including substantiation of the choice of the number of new-item copies.

Planning of Truncated Sequential Binomial Test via Relative Efficiency

A planning methodology is proposed for the sequential probability ratio test (SPRT) for the purpose of practical application. The SPRT is the most common acceptance test in the field of reliability and quality control. In it, the hypothesis is checked that the percentage of defective items does not exceed a specified value. Truncation is resorted to compensate for the absence of a limit on the test duration, but it complicates the planning process. Moreover, the discreteness and multidimensionality of the characteristics of such tests prevent their direct comparison and optimization. To remedy these drawbacks, quality features of the test are proposed, one of which—the relative efficiency—represents the ratio of the tests weighted average sample number till its stopping and its counterpart for the nontruncated SPRT. It facilitates solution of the problems in automatic planning of the test. Another important advantage of this relative efficiency is that it yields accurate and simple formulas for the stopping boundary. Besides, these formulas permit sound choice of the truncation level already at early stages of the planning process. A planners algorithm and an industrial example are also included. The proposed methodology can also be applied to exponential SPRT. The advantages of tests based on the proposed methodology over those in IEC-61123 (the binomial case) and IEC-61124 (the exponential case) are demonstrated, and revision of the standards is recommended. Copyright

Sequential testing for two exponential distributions at arbitrary risks

Purpose – This paper aims to propose a methodology for planning of a truncated sequential probability ratio test (SPRT) in which two systems with exponentially distributed times between failures (TBFs) are compared. The study is concerned with tests with arbitrary probabilities of I‐ and II‐type errors.Design/methodology/approach – The study methodology, based on the proposed optimality criteria for these tests, permitted comparison of different modes of truncation and obviated the drawbacks of discreteness and multidimensionality of their characteristics.Findings – The solution permits planning of a heavily‐truncated test with an average sample number exceeding its counterpart for the optimal (non‐truncated) test by at most a specified percentage. Relationships are outlined for optimal selection of the truncated test boundaries. So are optimality estimation criteria for the constructed test. The superiority of the SPRTs, truncated by the proposed methodology, over their counterparts, processed according ...

Improvement on “Sequential Testing” in MIL-HDBK-781A and IEC 61124

This paper presents results of an analysis of the sequential test (ST) procedures described in MIL-HDBK-781A, and IEC 61124, intended for checking the mean Time Between Failures (TBF) value under an exponential distribution of the TBF. The methodological basis of the calculations consists in discretization of the ST process through subdivision of the time axis in small segments. By this means, the process is converted into a binomial for which an algorithm, and a fast computer program have been developed; and most important of all, a tool is provided for searching for the optimal truncation. The influence of truncation by time on the Expected Test Time (ETT) characteristics was studied; and an improved truncation method, minimizing this influence, was developed. The distributions of the test times were determined. The type A plan characteristics in IEC 61124:2006 have substantial inconsistencies in the probabilities of types I & II errors (up to a factor of 2), and in the ETT (up to 17%). We checked these results by using the binomial-recursive method, and by simulation. The Type C plans, reproduced from GOST R27.402:2005, are consistent; but there is scope (and need) for substantial improvement of the search algorithm for the optimal parameters.

Ignatova, I., Deutsch, R. C., and Edwards, D. (2012), “Closed Sequential and Multistage Inference on Binary Responses With or Without Replacement,” The American Statistician, 66, 163-172: Comment by Michlin and Shaham and Reply

Editor’s Note: The letter by Segal and Torres inspired quite a bit of comment. Alas, we do not feel that TAS is the appropriate venue for continuing this discussion. However, we point out to our readership (as was pointed out to us by Raj Govindaraju) the comments made on this issue in the article and discussion of F. J. Anscombe’s (1967) paper, ‘‘Topics in the Investigation of Linear Relations Fitted by the Method of Least Squares” Journal of the Royal Statistical Society, Series B, 29, 1–52.

Sequential Test for Reliability Under Allowance for Target Uncertainty

This paper deals with development of a sequential test for comparison of the mean times between failures (MTBF) of two items: a new one (subscript n) and another serving as a reference (subscript ref). In the test, the hypothesis is checked that MTBFn is not less than a given multiple of MTBFref. Prior of the test, information is available on a known quantity r of the time between failures (TBF) for the reference item, which is not involved in the test, and only the new item is tested. It is a common occurrence in industrial practice that a new, modernized item has to be checked versus a reference counterpart currently in use. We assume that the TBF of both items is distributed exponentially. The sequential probability ratio test (SPRT) is widely used for checking that an MTBFn is not less than a specified (target) value. The innovation here is that in the proposed sequential test the target is the MTBFref which is available not as an exact value but only as an estimate whose uncertainty depend on r . Accordingly, a search algorithm was developed for the decision boundaries in the proposed test, as well as for calculating the operating characteristic (OC), and the statistical parameters of the test duration and number of failures to stopping the test. Approximative relationships were obtained for the above, depending on the target OC. These straightforward relationships drastically simplify the planning process. Moreover, they will enable the reliability engineer to appreciate the need to take the above uncertainty into account, and the expediency of resorting to the test at an early stage of its planning. Its superior efficacy is demonstrated.

Reliability improvement of Internet web servers through statistical process control

The proposed method, General Cross-Correlation Process Control (GCCPC), is intended for timely detection of hacker attacks and faults, before they become critical and disrupt the normal activity of a web server. It is based on observation of the correlations between the variables measured in the course of server activity. Some of the stronger of these correlations undergo drastic weakening at the onset of faults. A methodology is presented for choice of the variables to be observed, and of the monitoring parameters. To be noted are the robustness of the method, and the low cost of installation and maintenance of the system. Recommendations are given on real-world implementation of the method, as a basis for an automated protection system.

A proposal for updating the standard IEC-61123 - compliance test plans for success ratio

An improvement of IEC-61123 in the field of sequential probability ratio test (SPRT) is proposed. The current standard does not provide a proper solution for modern industrys needs, and the test plans are not up-to-date with the knowledge in the area of sequential tests. The advantages of the proposed version are reflected by the efficacy and accuracy of the tests, the wider range of the ready to use tests parameters, and available data regarding the tests characteristics. The proposed version is a significant improvement over the existing one. The changes will extend the use of SPRT and this standard.

Testing performance of IT service associates under variable load

One of the challenges in IT Service Management is an assessment of an innovation benefit introduced into service delivery on performance of IT System Administrator (SA). The question is especially complex when variability of the service requests load is taken into account. We used the sequential probability ratio test to assess effects of innovation for a case when a load of the service teams is constant over time. One of the drawbacks of this test is its requirement for a constant SAs team size while the arrival patterns of service requests might require otherwise. To address the volatility of a service request load we consider the possibility of changing number of service associates. We design a performance test based on the concepts of Repeated Significance Test (RST), which allowed us to take variable number of SAs into account. In this work we show the superiority of the suggested test in comparison to the fixed sample size test (FSST) in a number of experiments required to reach the definitive conclusion. We also compare RST with SPRT when the size of the service team is constant.

Planning of truncated sequential binomial tests via the ASN-increase parameter

The Sequential Probability Ratio Test (SPRT) is the most common acceptance test in the field of reliability and quality control of electronic systems. Proposed are measures of the test quality. One of them is the increase in the Average Sample Number (ASN) caused by the test truncation. An optimality criterion based on it considerably facilitates solution of the problems in automatic planning of the test. Also given are formulas for determining one of the parameters of the test boundaries — the Truncation Apex (TA), depending on the required test characteristics. A users algorithm for the planner is also included.