Operating Characteristic Curves for the Exponential Bayes‐Truncated Test

Operating characteristic (OC) curves are often useful in determining how large a sample is required to detect a specified difference for a particular consumer and producer risk. In this paper, OC curves with Bayes stopping rules for the exponential distribution are developed. Example curves are provided for the sequential and batch testing situations. The power of the test is greater under batch testing. A table illustrates the performance of the plans for stopping at each opportunity in a sample of size 20. Some examples are used to demonstrate the application of the proposed methodology. Copyright © 2004 John Wiley & Sons, Ltd.     

A sequential hypothesis testing procedure for the process capability index Cpk

In this study, we propose a sequential method for hypothesis testing on the C_pk process capability index. We compare the properties of the sequential test with the performances of nonsequential tests by performing an extensive simulation study. The results show that the proposed sequential test makes it possible to save a large amount of sample size as compared with the fixed sample size tests while maintaining the desired α‐level and power.     

A Classification of Fallacious Arguments and Interpretations

This paper describes from first principles the direct calculation of the operating characteristic function, O.C., the probability of accepting the hypothesis θ = θ₀, and the average sample size, A.S.N., required to terminate the test, for any truncated sequential test once the acceptance, rejection, and the continuation regions are specified at each stage. What is needed is to regard a sequential test as a step by step random walk, which is a Markov chain. The method is contrasted with Wald's and two examples are included.     

The Distribution of Observed Defectives in Attribute Acceptance Sampling Plans Under Inspection Error

The Type B Operating Characteristic curve in attributes acceptance sampling is simply the probability of lot acceptance versus process fraction defective. The binomial mass function is appropriate for assessing this acceptance probability. Articles on inspection error in attributes acceptance sampling frequently assume the marginal distribution of the number of observed defectives in the sample is binomial with the true fraction defective replaced by the apparent fraction defective. This assumption, however, has been questioned and it is the object of this paper to show explicitly that the existing literature is quite correct. Following this development, additional results are presented which may be useful to others performing research in this area.     

Introduction To Probability and Statistical Decision Theory

A nonparametric sequential rank correlation test for independence consisting of a generalized sequential probability ratio test is investigated through the Monte Carlo method. The test is found to be satisfactory in terms of the power function. Comparison of the expected sample size of the test and three other tests including the fixed sample size test suggests that the sequential rank test is also efficient.     

Acceptance sampling based on reliability degradation data

Degradation test is often performed in order to ensure high reliability of an industrial product at the manufacturing stage. Typically, a test is conducted by taking a sample. Therefore, adopting an effective acceptance sampling plan is an important issue. However, reliability literature scarcely dealt with such a topic. The main purpose of this paper is to design a single sampling plan for a lot of industrial products in lieu of degradation test. We consider both fixed and random effects models to describe degradation patterns of samples. Proposed procedures are applied to the case of keyboard degradation test.     

A Two-Sample Test for Reliability Comparison

A new test for the comparison of reliabilities of two populations is proposed. The test statistic uses only the number of failures in each sample. The test procedure is conditional in the sense that the unknown parameter p, the probability of failure during the test period, under the null hypothesis is estimated first by the pooled sample proportion. Critical values are tabulated for several equal sample size cases and asymptotic properties are obtained. The use of this test in practice is illustrated by several examples. It is shown that the power of the proposed test is larger than that of Fisher's exact test for the cases considered. Applications in survival analysis are also studied, and the power of the test procedure is compared with those of other tests, which are based on failure times in addition to failure counts.     

Sequential Testing of Sample Size

A sample of unknown size n is obtained from a known population. The order statistics are then examined sequentially for the purpose of drawing inferences about n. A procedure for testing the unknown sample size n is given and its exact properties are obtained. This procedure provides a basis for examining some popular conjectures and approximations in sequential analysis. In particular two suggested approximations to the average sample number of a sequential probability ratio test are discussed.     

Evaluating and Implementing 3-Level Acceptance Sampling Plans

There are many situations in which product quality can be described by classifying a product using three or more discrete levels. For example, a food product may be classified as good, marginal, or bad depending on the concentration of harmful microorganisms in the product. In this paper, a generic framework is defined for establishing 3-level acceptance sampling plans. These plans utilize what we refer to as quality value functions. The Operating Characteristic function for these plans is constructed and used to develop an approximate parameter selection method based on the Central Limit Theorem. The results of testing this method using numerical examples are presented. The problem of quality value function selection is also addressed. A detailed example is presented, which includes the implementation of both the parameter and quality value function selection methods.     

A Truncated Sequential t-Test

A truncated sequential t-test is developed and is compared with Barnard's one-sided (one-tailed) non-truncated sequential t-test and the fixed sample-size procedure. Empirical results indicate that, in part of the parametric region between the null and alternative hypotheses, the proposed test, as compared to Barnard's procedure, leads to improvement in ASN and appreciable reduction in sample size variability. In addition, it seems to have uniformly lower ASN Values than those of its fixedsample counterpart. The proposed test is then extended to a two-sided procedure and compared with the truncated t-test of Schneiderman and Armitage.     

Planning of sequential binomial tests for various risk ratios

The Sequential Probability Ratio Test (SPRT) is widely used in the field of reliability and quality control. This paper is a continuation and a significant extension of the authors' earlier paper; it is dedicated to various risk ratios (α/β) and will lead to the increased use of the Sequential Probability Ratio Test for practical and research needs. The sample number (SN) until the test stops is a random value, and its distribution tails can be extremely long relative to the average SN (ASN). This is not suitable for practical use; therefore, truncation is required, usually by a pair of lines whose intersection, denoted as the Truncation Apex (TA), determines the maximum SN (maxSN). The optimality of the test is determined by the minimality of the SN (by means of maxSN and ASN) for a given Operating Characteristic. Presented are formulas and an algorithm for the TA and other parameters of the optimal test stopping boundaries for various α/β. This methodology also shortens the test planning process. Displacement of the TA from the optimal location results in a significant increase in ASN. The study was implemented in the Israeli standard SI‐61123. Revision of IEC 61123 and IEC 61124 (for exponential distributed data), by this study, has been accepted to the work plan of TC‐56 of IEC. The proposed methodology can be the basis for the improvement of additional standards, for example, in ISO 8422:2006.     

An Elementary Introduction to the Theory of Probability

This paper describes modifications of the sequential, two-sample, grouped, rank tests developed earlier. A modified, sequential, configural rank test is discussed in some detail and is a procedure based on rerankings of observations as new groups of observations are obtained sequentially. A numerical application is given.

Monte Carlo results are presented on the modified, sequential, configural rank test and compared with earlier studies. Proofs of termination of the test are reported and outstanding unsolved problems noted.     

Parametric Estimation

The exponential distribution is often used in reliability work to describe the distribution of time to “chance” failure and is characterized by a constant failure rate. In this paper the small sample powers are compared for four test statistics for the hypothesis of constant failure rate vs. the hypothesis of non-constant failure rate. The tests are compared for samples of size n = 10(5)50 using the Weibull distribution for the alternative distribution. The shape parameter of the Weibull is varied from 0.5 to 2.5. For the two test statistics which involve arbitrary grouping of the data the effect of group size and number was also examined.     

Inspection Error Adjustment in the Design of Single Sampling Attributes Plan

This article presents an approach for the design of single sampling attributes plans of given strength when inspection errors are present. The design criterion considered is the specification of two points of the Operating Characteristic curve. The design approach also recommends repeated testing of sampled nonconforming units for conformance in order to reduce the adverse effect of the inspection errors. It is shown that realistic sampling plan parameters can be obtained if the inspection error probabilities are taken into account in the design of sampling plans.     

Optimal Acceptance Test Plans with Grouping

This paper considers the problem of selecting an optimal acceptance test, plan. A cost model is developed which permits comparison of single, dollble, multiple, and sequential test plans. Cost elements include fixed costs to initiate testing, costs to purchase and ship components, and variable costs for testing and return or disposal of untested items.

In the case of seqllential testing, consideration of these cost factors leads to an optimization model having as decision variables the number of samples to ship to the test facility for each possible round of tests. Solution of this optimization problenl enables explicit comparison of test costs with non-sequential designs. A numerical example is provided for illtlstration.     

SEQUENTIAL WORK SAMPLING TESTS

In this paper a test method for time study developed from a theoretical basis of sequential analysis of mathematical statistics is presented. This method admits a statistical decision on the regions which contain a certain time ratio.

In the course of the test process, the instantaneous observations of the activities made at the work places are assigned by small integers which yield a simple sequential statistic. The observations need not be recorded. The sample size is, on the average, considerably smaller than that of comparable studies by normal work sampling.

The design and application of sequential work sampling tests are described with regard to the estimation method of work sampling and their most important features are discussed.     

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 test's 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 planner's 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 © 2012 John Wiley & Sons, Ltd.     

Point and interval estimation of accuracies of a binary medical diagnostic test following group sequential testing

When hypotheses concerning the sensitivity and specificity of a binary medical diagnostic test are simultaneously tested using a group sequential procedure, constructing point and interval estimates of the parameters is challenging because there is no unique way to order sample points in the two-dimensional space. In this paper, upon termination of a group sequential procedure, we compare the bias and mean squared errors of the maximum-likelihood and Rao-Blackwell unbiased estimators of sensitivity and specificity. Confidence intervals (CIs) of the two parameters were constructed using normal approximation and Woodroofe's pivot methods based on maximum-likelihood and Rao-Blackwell unbiased estimates. The coverage probability and the expected length of CIs for the parameters were compared by simulation studies.     

Acoustic microscopy: a tool for non-destructive evaluation of ceramics

The focusing properties of the lenses used in the scanning acoustic microscope (SAM) have been widely studied for the particular case of surface imaging. When focusing into the bulk of a sample the aberration of the lens become negligible with respect to that induced by the sample surface traversal. A more detailed study allows evaluation of the acoustic field distribution, using an angular spectrum expansion. This formalism, previously developed by several independent authors, is used for optimizing the geometrical parameters of the acoustic lens for non-destructive testing of ceramic materials. The deduced theoretical performances are compared with experimental findings at a working frequency of 100 MHz. Characteristic images for some ceramic materials are presented, which give valuable information about the geometry of the defects and the structure of the matrix material.A comparison with alternative non-destructive testing methods is also given.     

Tests for One or Two Outliers in Normal Samples with Unknown Variance: A Correction

In a paper in this Journal, McMillan (1971) compared three procedures for the detection of outliers in samples from a Normal population. One of these was a sequential application of a maximum residual test. One step in McMillan's derivation of the probability of detecting two outliers using this test is incorrect. A correct derivation is presented in this note and some numerical results in McMillan's paper are appropriately amended. McMillan's qualitative statements on the performance of the test are essentially unaffected.