Showing papers on "Sequential probability ratio test published in 1975"

Optimal allocation in sequential tests comparing the means of two Gaussian populations

Abstract: SUMMARY The invariant sequential probability ratio test used in testing for a difference between the means of two Gaussian populations is set up. The error probabilities for this test are effectively constant over a rich class of data-dependent allocation rules. The additional risk, average sample number plus (y - 1) times the expected number of observations to the inferior population, for y > 1, is introduced and the optimal allocation rule is found for the continuous-time analogue to this problem. Analytical results show this rule to be asymptotically optimal in discrete time, and simulations indicate its near optimal per- formance for the finite case. The problem of two-population hypothesis testing with data-dependent allocation of observations has been treated by several authors. Also, the applications of this decision model, especially to clinical testing, have been well documented. Recent results show that when the test is sequential and the termination rule is of the sequential probability ratio test type, the probability of correct hypothesis selection is constant, ignoring overshoot, for a rich class of data-dependent allocation rules; see Flehinger, Louis, Robbins & Singer (1972) and an as yet unpublished paper of mine. This constancy permits one to search the class for a rule which performs well with respect to some additional cost structure, one usually based on the number of observations taken on the superior and inferior populations. Flehinger & Louis (1972) and Robbins & Siegmund (1974) give simulations showing that a substantial reduction in the expected number of observations on the inferior popu- lation is possible using data-dependent allocation rules, as opposed to equal assignment, for the case of comparing two Gaussian populations with known variances. The simulation results of Flehinger & Louis (1971) show the same reduction for the exponential distri- bution. In the present paper the risk function formed from the average sample number plus (y - 1) x the expected number of observations allocated to the inferior population, for y > 1, is introduced into the Gaussian testing model. Here y is the relative cost of taking an observation from the inferior as opposed to the superior population, and varying y allows one to balance the two components of risk. In ? 2 first the Gaussian allocation and testing problem is set up and previous results are summarized. Using the above risk function, in Appendix A the optimal allocation and its risk are obtained for the continuous-time idealization, that of comparing the drifts of two Brownian motions. Back in ? 2 this optimal rule is related to the discrete testing situation.

Locally most powerful sequential tests

Abstract: 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.

On Some Parametric and Nonparametric Sequential Subset Selection Procedures

Abstract: Publisher Summary This chapter discusses some parametric and nonparametric sequential subset selection procedures. In sequential selection and ranking procedures, the best population is selected by using the indifference zone approach. Multiple decision procedures can be defined that select a subset of a fixed or random size. The chapter describes a method to construct nonelimination type sequential procedures to select a subset containing all the superior populations. These procedures are formed by choosing a statistic based on the observations from any two of the populations and performing a sequential probability ratio test (SPRT) based on this statistic. The selection procedure is applied to the problem of choosing normal populations for the superior populations with smaller variance. An appropriate population is removed from a set of contending populations, if the associated SPRTs terminate. This is continued until the elimination is stopped.

An Unbiased Estimator and a Sequential Test for the Correlation Coefficient

Abstract: An unbiased estimator and a Sequential Probability Ratio Test (SPRT) for the correlation coefficient are proposed. The SPRT reduces to the SPRT for a binomial parameter. A comparison is made of the proposed test with those of Kowalski [4] and Choi [1].

A Sequential Nonparametric Pattern Classification Algorithm Based on the Wald SPRT

Abstract: A sequential nonparametric pattern classification procedure is presented. The method presented is an estimated version of the Wald sequential probability ratio test (SPRT). This method utilizes density function estimates, and the density estimate used is discussed, including a proof of convergence in probability of the estimate to the true density function. The classification procedure proposed makes use of the theory of order statistics, and estimates of the probabilities of misclassification are given. The procedure was tested on discriminating between two classes of Gaussian samples and on discriminating between two kinds of electroencephalogram (EEG) responses.

A Modified Play-the-Winner Rule for Sequential Trials

Abstract: The two-arm bandit type of Play-the-Winner rule is modified to sample simultaneously the two competing sequences of trials until the first “failure” occurs. Then the sequence without a failure is sampled until it fails, and the process is repeated. This modification leads to the direct application of the sequential probability ratio test on two paired geometric variables. A further modification leads to a scheme which avoids overshooting the stopping boundaries and reduces the total sample size. Exact formulas for the boundary limits, operating characteristics and ASN function for the proposed methods are presented.

The Sequential Probability Ratio Test

Abstract: During World War II, Abraham Wald started working on sequential procedures and he developed the sequential probability ratio test procedure. Neyman and Pearson have provided a method of constructing the most powerful test for a simple hypothesis versus simple alternative-testing problem. It has been shown that although one might obtain a sequence of observations that called for continuing the sampling at each stage, with no decision reached, the probability of such an occurrence is zero. The sample size needed to reach a decision in a sequential or a multiple sampling plan is a random variable. The distribution of this random variable depends on the distribution that actually obtains during the sampling process, that is, on the state of nature.

A sequential median test with applications to CEP testing

Abstract: : A sequential test of a simple hypothesis of the distribution of a random variable against a simple alternate hypothesis is proposed The test terminates as soon as one of a sequence of sequentially observed sample medians falls outside a 'continuation region' The test can also be used for hypotheses concerning the median of the sampled population, and is especially useful when hypothesized distributions may provide poor fit in the tails, in which case 'outliers' may seriously degrade the performance of traditional procedures such as the Sequential Probability Ratio Test Applications to testing hypotheses about the circular error probable of weapon systems are discussed, and the tables of stopping bounds for such tests are presented (Author)