Milton Sobel

Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas

For a sequence of independent Bernoulli trials with probabilities of success and failure p and q respectively, the sooner and later waiting time random variables for the succession quota (s successes and r failures) is studied by deriving a generalized pgf and obtaining the means and variances. The same problems are considered with the usual frequency quota (s successes and r failures) using Dirichlet integral methodology; the above results hold for general p (0

Binomial Group-Testing With an Unknown Proportion of Defectives*

The binomial group-testing problem is extended to the case in which the common probability p of a unit being defective is unknown. A Bayes “non-mixing” procedure R (1) is derived and compared with other procedures, in particular with the corresponding procedure R 1 that requires the knowledge of p and with another procedure based on continually revising the maximum likelihood estimate of p; the latter is called an empirical Bayes solution. Finally, a Bayes procedure derived in the Appendix allows “mixing” and this is conjectured to be the unrestricted Bayes solution; the improvements due to “mixing” are shown to be small in Table III. Several applications of the general problem of group-testing are discussed in the introduction and this virtually constitutes a general review of the subject. After the procedure R (1) is defined a detailed illustration is given in section 2.1 showing how to carry out this procedure with the use of Tables I and II. Lower bounds for the Bayes risk associated with any group te...

A Survey of Adaptive Sampling for Clinical Trials

Publisher Summary This chapter reviews some work on a subject that treats the design of clinical trials in the light of ranking and selection theory. Although adaptive sampling rules have a logical and mathematical appeal, equal allocation plans often have the virtue that patients can be matched with respect to relevant biological factors. There are, however, many trials in which patients enter sequentially in time and detailed matching is either not possible or not important. All the theories employ a single measure of what constitutes a better population. Very often, the physician is called on to balance several factors, with two obvious examples being cost and side effects. There has been no theoretical examination of such vector criteria, and there has been little formal work done on balancing different factors so that they can be included within a single theoretical framework. In a case presented in the chapter, it is shown that a two-stage procedure requires only a small number of preliminary tests to benefit from the possibility of choosing between allocation methods. Too little has been done on clinical trials in which the measured parameter is a patients lifetime. The possibility of switching treatments on a patient for whom it is found that the initial treatment is leading to failure is important. This strategy is often used in cancer chemotherapy, but little is available by way of guidance to clinicians from theory. Many more allocation and stopping rules should be examined in light of the objective of reducing the number of patients on relatively poor treatments and on designing simple and efficient clinical trials.

Play-the-Winner Rule and Inverse Sampling in Selecting the Better of Two Binomial Populations

Abstract In this article two sampling rules are analyzed for the problem of selecting the better of two binomial populations. The first is alternate sampling (vector-at-a-time) and the second is the “play-the-winner” rule introduced by Robbins [3], and discussed by Zelen [8]. The termination rule studied is inverse sampling. One result is that the probabilities of correct selection for these two methods of sampling are exactly equal. It is also found that the play-the-winner sampling is uniformly preferable to the vector-at-a-time sampling rule in the sense that for the same probability requirement the expected number of trials on the poorer population is always smaller.

On Partitioning a Sample with Binary-Type Questions in Lieu of Collecting Observations

Abstract The problem is to search for the t largest observations in a random sample of size n by asking binary-type questions of the people (or items) in the sample without collecting any exact data whatever. The unordered and ordered cases are both considered; in the latter case the complete ranking is of special interest. Two different criteria of optimality are considered: (a) to minimize the expected number of questions required and (b) to maximize the probability of terminating the search in at most r questions for specified r. Optimal procedures are found and compared; in some sense the solutions for these two criteria are close to each other. The analysis is nonparametric in the sense that it holds for any underlying sampling distribution, but the actual optimal procedures depend on the specified distribution. In the above, we count (the cost of) a question as one regardless of the number of people addressed; other models in which the cost depends on the number of people are considered only briefly...

Finding a Single Defective in Binomial Group-Testing

Abstract The problem of finding a single defective item from an infinite binomial population is considered when group-testing is possible, i.e., when we can test any number of units x simultaneously and find out in one test if all x are good or if at least one of the x is defective. An optimal procedure is obtained in the sense that it minimizes the expected number of tests required to find one defective. Upper and lower bounds are derived using information theory and the relation of our procedure to the Huffman algorithm and the corresponding cost is studied.

Incomplete Dirichlet integrals with applications to ordered uniform spacings

Sobel, Uppuluri, and Frankowski, (Selected Tables in Math. Statistics, Vol. IV. Amer. Math. Soc., Providence, R. I., 1977) consider an incomplete Dirichlet integral of type I with several interesting applications connected with the multinomial distribution and provide tables of this integral along with other useful tables. Two incomplete Dirichlet integrals are discussed here along with some useful recurrence relations, providing simple methods of deriving the distribution theory of ordered uniform spacings.

A subset selection technique for scoring items on a multiple choice test

On a multiple-choice test in which each item hask alternative responses, the test taker is permitted to choose any subset which he believes contains the one correct answer. A scoring system is devised that depends on the size of the subset and on whether or not the correct answer is eliminated. The mean and variance of the score per item are obtained. Methods are derived for determining the total number of items that should be included on the test so that the average score on all items can be regarded as a good measure of the subjects knowledge. Efficiency comparisons between conventional and the subset selection scoring procedures are made. The analogous problem ofr > 1 correct answers for each item (withr fixed and known) is also considered.

Quota sampling for multinomial via Dirichlet

Abstract For a sequence of independent multinomial trials with known cell probabilities, an inverse-sampling procedure is studied that is based on stopping when any one of several cell quotas is satisfied; each quota is associated with a different cell and we are dealing with the ‘soonest’ problem for these quotas. Certain cells have frequency quotas (i.e., based on the total frequency accumulated in that cell); other cells disjoint from these have run quotas (i.e., based on the length of the current run in that cell), and we may have at most one ‘slack’ cell without any quota. If there are α cells with frequency quotas, β cells with run quotas and one slack cell, we call it Case (α,β,1). The goal is to find the expected waiting time as well as the variance of the waiting time needed to reach any stopping point. Extensive explicit formulas are given for the Case (1, 1, 0) and other cases are illustrated in the text. The Case (0,β,0) and Case (0,β,1) are related to some work of W. Feller; the present paper can be regarded as a generalization of some of his work both in the direction of combining frequency quotas and run quotas in the same problem, and in adding more cells.

An Integrated Formulation For Selecting The t Best Of k Normal Populations

We refer to the two classical approaches to ranking and selection as the indifference zone approach and the subset selection approach. This paper integrates these two appraches by separating the parameter space into two distionit parts, the preference zone (PZ) and the indifference zone (IZ). In the PZ we insist on selecting the t best for a correct selection (CS1) but in the IZ we define any selected. We then use different methods to find two constants and a common sample size n that simultaneously give lower bounds P1 for (CS1PZ) and Po for P(CS2IZ). Here the values of P1. Po snf (which defines the PZ) are all specified and can be arbitraily close to 11 and 0, respectively. Explicit results are given for the P(CS), E(S), P(S=k) and P(S=t), especially for the slippage configuration (SPC) and the equal parameter configuratim (EPC). It is shown that the former is least favorable in the PZ and, for t = l, that the latter is the worst case in the IZ. An illustrative exmple is included, but extensive tables h...