Wald's SPRT Research Articles

A generalized Shiryayev sequential probability ratio test for change detection and isolation

The authors derive an online multiple hypothesis Shiryayev sequential probability ratio test (SSPRT) by adopting a dynamic programming approach. It is shown that for a certain criterion of optimality, this generalized Shiryayev SPRT detects and isolates a change in hypothesis in the conditionally independent measurement sequence in minimum time, unlike the Wald SPRT, which assumes the entire measurement sequence to correspond to a single hypothesis. The measurement cost, the cost of a false alarm, and the cost of a miss-alarm are considered in our dynamic programming analysis. The algorithm is shown to be optimal in the infinite time case. Finally, the performance of the algorithm is evaluated by using a few examples. In particular, they implement the algorithm in a fault detection and identification scheme for advanced vehicle control systems.

A generic grouping algorithm and its quantitative analysis

This paper presents a generic method for perceptual grouping and an analysis of its expected grouping quality. The grouping method is fairly general: It may be used for the grouping of various types of data features, and to incorporate different grouping cues operating over feature sets of different sizes. The proposed method is divided into two parts: constructing a graph representation of the available perceptual grouping evidence, and then finding the partition of the graph into groups. The first stage includes a cue enhancement procedure, which integrates the information available from multifeature cues into very reliable bifeature cues. Both stages are implemented using known statistical tools such as Wald's SPRT algorithm (1952) and the maximum likelihood criterion. The accompanying theoretical analysis of this grouping criterion quantifies intuitive expectations and predicts that the expected grouping quality increases with cue reliability. It also shows that investing more computational effort in the grouping algorithm leads to better grouping results. This analysis, which quantifies the grouping power of the maximum likelihood criterion, is independent of the grouping domain. To our best knowledge, such an analysis of a grouping process is given here for the first time. Three grouping algorithms, in three different domains, are synthesized as instances of the generic method. They demonstrate the applicability and generality of this grouping method.

A Class of Sequential Conditional Probability Ratio Tests

Abstract A new class of sequential testing procedures, sequential conditional probability ratio tests (SCPRT's), is proposed. These procedures have the property of having a maximum sample size no larger than a fixed-sample test with an almost identical power function. The efficiency of the SCPRT, in terms of expected sample size, is close to that of Wald's SPRT. Group sequential testing procedures derived from the SCPRT are not restricted to the choices of group sizes and number of groups. At the same time, the SCPRT, the group-SCPRT, and the reference fixed sample size test (RFSST) are agreeable in the sense of having very small probability of leading to different rejection/acceptance results. For the SCPRT or the group-SCPRT, stopping boundaries are decided by appropriate deflection factors determined by controlling absorption probabilities of a random bridge on the SCPRT boundaries. In this article the basic idea for the SCPRT is introduced in a general setup. But examples and technical discussions are restricted mainly to dichotomous distributions, in which the power functions and expected sample sizes for the sequential procedures can be easily computed by a method developed by Xiong.

Computation of Some Exact Properties of Wald's Sprt when Sampling from a Class of Discrete Distributions

AbstractAn algorithm is developed for determining exact values of the Operating Characteristic and Average Sample Number functions for Wald's Sequential Probability Ratio Test (SPRT) when observations are drawn from a discrete, Koopman‐Darmois density with positive probability on the nonnegative integers. The sample size distribution and percentiles of sample size are also considered. An example is given for the negative binomial distribution.

Locally most powerful sequential tests for processes of the exponential class with stationary and independent increments

Sequential tests that are locally most powerful (LMP) for certain one-sided problems in the case of processes of the exponential class with stationary and independent increments are discussed. After having proved the existence of an LMP sequential test in the above setting each LMP test is shown to be a Wald SPRT for a family of paired (conjugate) simple hypotheses.

Optimality Of sequential prorability ratio tests for a class of continuous parameter stochastic processes

For a wide class of stochastic processes including processes belonging to the curved exponential families, it is proved that Wald SPRT is optimal in the sense of minimizing the expectations of increasing processes associated with the stochastic process. The increasing processes are given by the Doob-Meyer decompositions of the log-probability ratio under the two hypotheses.

An upper bound for the accuracy of the Wiener process approximation of the error probabilities of the sprt

For Wald's SPRT of a simple hypothesis against a simple alternative an upper bound for the differences of the error probabilities and their Wiener process approximations is derived. This upper bound only depends on the first three moments of the log-likelihood ratio and is seen to be especially useful in the case of contiguous hypotheses.

A general inequality for the rejection probability of the sprt

Let be i.i.d. random variables and SO that N is the stopping rule of Wald's SPRT, if the Zi are log-likelihood statistics. For the case that and an upper bound of order for is derived.

Asymptotic Optimality of Invariant Sequential Probability Ratio Tests

It is well known that Wald's SPRT for testing simple hypotheses based on i.i.d. observations minimizes the expected sample size both under the null and under the alternative hypotheses among all tests with the same or smaller error probabilities and with finite expected sample sizes under the two hypotheses. In this paper it is shown that this optimum property can be extended, at least asymptotically as the error probabilities tend to 0, to invariant SPRTs like the sequential $t$-test, the Savage-Sethuraman sequential rank-order test, etc. In fact, not only do these invariant SPRTs asymptotically minimize the expected sample size, but they also asymptotically minimize all the moments of the sample size distribution among all invariant tests with the same or smaller error probabilities. Modifications of these invariant SPRTs to asymptotically minimize the moments of the sample size at an intermediate parameter are also considered.

A modification of the partial sequential probability ratio test

For the problem of discriminating between two simple hypoth¬eses concerning a Koopman - Darmois parameter, a modification of the partial sequential probability ratio test is proposed where instead of drawing only one fixed sample, two fixed samples are drawn and then Wald's SPRT is started. The OC and the ASN func¬tions are derived. Numerical comparisons are made with Wald's and Read's procedures for testing the normal mean with known variance. For some parameter values, the test procedure has a lower ASN than that of Read's procedure.

Asymptotic Efficiencies of Sequential Tests

Some concepts of relative and absolute efficiency for sequential tests are considered. These are sequential analogs of Hodges-Lehmann and Chernoff efficiencies. Using these criteria, several sequential tests for the mean of a normal distribution are evaluated. Among them are Wald's SPRT, Anderson's triangular boundary, Bayes and APO tests and a repeated significance test. Truncated versions of these tests are also considered. Asymptotic expressions for the (expected) stopping times and error rates are given.

Two-Stage Plans Compared with Fixed-Sample-Size and Wald SPRT Plans

Abstract Within the Neyman-Pearson framework of hypothesis testing with fixed-error-level specifications, two-stage designs are obtained such that sample size is minimized when the alternative hypothesis is true. Normally distributed variates with known variance and binomially distributed variates are considered. It is shown that when the alternative hypothesis is true, these optimal two-stage designs generally achieve between one-half and two-thirds of the ASN differential between the two extremes of analogous fixed-sample designs (maximum ASN) and item-by-item Wald SPRT design (minimum ASN when alternative hypothesis is true).

Two-Stage Plans Compared with Fixed-Sample-Size and Wald SPRT Plans

Abstract Within the Neyman-Pearson framework of hypothesis testing with fixed-error-level specifications, two-stage designs are obtained such that sample size is minimized when the alternative hypothesis is true. Normally distributed variates with known variance and binomially distributed variates are considered. It is shown that when the alternative hypothesis is true, these optimal two-stage designs generally achieve between one-half and two-thirds of the ASN differential between the two extremes of analogous fixed-sample designs (maximum ASN) and item-by-item Wald SPRT design (minimum ASN when alternative hypothesis is true).

Termination, Moments and Exponential Boundedness of the Stopping Rule for Certain Invariant Sequential Probability Ratio Tests

It is well known that Wald's SPRT terminates with probability one and in fact the stopping time is exponentially bounded for every distribution $P$ except in the trivial case where the $\log$ likelihood ratio vanishes with probability one. The results in the literature for invariant SPRT's, however, have been considerably less complete. In all the parametric problems studied, moment conditions of certain random variables have been assumed to prove termination, and the finiteness of their moment generating function has also been assumed to prove the exponential boundedness of the stopping rule. In this paper, we try to remove or weaken these conditions for certain invariant SPRT's. In particular, we show that like the Wald SPRT, the sequential $t$- and $F$-tests always terminate with probability one for any distribution $P$ except in trivial cases. However, the stopping rules may fail to be exponentially bounded, and obstructive distributions are also exhibited. Sufficient conditions for exponential boundedness and finiteness of moments of the stopping rule are studied, and asymptotic expressions for the moments of the stopping rule are also given.

Locally Most Powerful Sequential Tests

are discussed. In all cases considered, the stopping rule is the first time a certain random walk leaves a bounded interval. (Thus various inequalities and approximations due to Wald can be utilized in obtaining properties of these tests.) For models in a one-parameter exponential family, each LMP sequential test is shown to be a Wald SPRT for a family of paired (conjugate) simple hypotheses.

Some Asymptotic Aspects of Sequential Analysis

The asymptotic behavior is given for the error rates and ASN of the Wald SPRT and of invariant sequential tests. An asymptotic justification of Bhate's conjecture is also provided for invariant sequential tests. Expressions are obtained for the asymptotic relative efficiency of the Wald SPRT as compared with the corresponding best non-sequential test.

The OC and ASN Functions of Some SPRT's for the Correlation Coefficient

The use of Wald's SPRT to test hypotheses concerning the correlation coefficient is complicated by the nuisance parameters (μ x , μ y , σ X , σ y ) in the bivariate normal density function. While more general theories have been proposed to produce sequential tests in such situations, the properties of these tests are difficult to ascertain since Wald's method of approximating the OC and ASN functions (usually) breaks down. We derive several sequential tests for the correlation coefficient for which these functions are calculable.

The OC and ASN Functions of Some SPRT's for the Correlation Coefficient

The use of Wald's SPRT to test hypotheses concerning the correlation coefficient is complicated by the nuisance parameters (μ x , μ y , σ X , σ y ) in the bivariate normal density function. While more general theories have been proposed to produce sequential tests in such situations, the properties of these tests are difficult to ascertain since Wald's method of approximating the OC and ASN functions (usually) breaks down. We derive several sequential tests for the correlation coefficient for which these functions are calculable.

The Partial Sequential Probability Ratio Test

Abstract In testing a normal mean with known variance, or a Koopman-Darmois parameter, an initial fixed number n of observations is followed by Wald's SPRT procedure. The conditional SPRT, given, and optimality properties in a certain class of tests are noted. For some parameter values, the PSPRT may have a lower ASN than a Wald SPRT with the same error probabilities.

The Partial Sequential Probability Ratio Test

Abstract In testing a normal mean with known variance, or a Koopman-Darmois parameter, an initial fixed number n of observations is followed by Wald's SPRT procedure. The conditional SPRT, given, and optimality properties in a certain class of tests are noted. For some parameter values, the PSPRT may have a lower ASN than a Wald SPRT with the same error probabilities.