Sequential Estimation and Closed Sequential Decision Procedures

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In Chapter 10 of his classical work on sequential analysis [8], Wald started a program for dealing with multiple decision problems based on sequential estimation, and this work was to some extent further developed by Stein [7]. However, as far as the present writer is aware, this program was never completed. In a recent paper [6] a sequential procedure for estimating the mean of a normal distribution was given and the results applied to the problem of deciding which of $k$ non-overlapping intervals contains the mean. In the present paper, the main results of [6] are first derived in a different manner, using a slight modification of a procedure given by Wald (see Chapter 10 of [8]). This new derivation lends itself to dealing with other situations, and sequential confidence limits are worked out explicitly for two other cases, namely, for the variance and for the ratio of variances of normal distributions. These results are then applied to get closed decision procedures for a number of decision problems, including (1) testing a hypothesis regarding the mean of a normal distribution against either one-sided or two-sided alternatives; (2) comparing the means of $k$ experimental categories with a standard or control; (3) testing a hypothesis about the variance of a normal distribution; (4) deciding which of $k$ non-overlapping intervals contains the variance; (5) testing a hypothesis about the ratio of variances; (6) comparing the variances of $k$ experimental categories with a standard or control. In addition, a brief discussion of mixed problems, where we are concerned with finding a confidence interval for the parameter after a decision has been reached is given in Sections 3.1 and 3.3. In some of these problems open sequential solutions (in which there is no upper bound to the number of measurements required to reach a decision) are already known which have some optimum properties. However, for administrative and other reasons it is often desirable to restrict consideration to closed sequential procedures, so as to have an upper bound to the time or cost of an experiment. This has recently been emphasized by Armitage in connection with medical applications [2]. All the sequential procedures of the present paper are closed, but the point of closure in Section 3 depends on $s^2$ if $\sigma^2$ is unknown.