A Problem In Sequential Analysis (stopping Time, Significance Level, Backward Induction).

Abstract

The problems addressed in this thesis are motivated by the following problem from clinical trials. Suppose that an experimental drug has been developed to treat a non-life-treatening disease, such as colds. Suppose that the drug is administered sequentially to incoming patients and that a response Y to treatment and a covariate X may be measured for each patient. For example, X might be a side effect. Suppose finally, that the experiment might be stopped if the covariate falls outside some acceptable range. We study the effect that such optimal stopping rules have on two conventional statistical measures; namely, the significance level and the bias. We present numerical solutions to the optimal stopping problems for both testing and estimation in Chapter 2 for some special cases. The effect of optimal stopping on the significance level is surprisingly small in the examples considered, an increase by less than a factor of two. Patients may be measured individually, but treated in batches of different sizes in both the case of hypothesis testing and estimation. In many situations treating in batches could be a more economical way of treatment. We consider this complication for both testing and estimation. From the data we obtain it seems that the size of the batch has a modest effect. In the theoretical analysis of the estimation problem in Chapter 3, we formulate an analogous problem in continuous time.The use of the Wiener process is a natural approximation, since we are concerned with the successive sums of independent observations to which the central limit theorem applies. For an appropriate normalization, we show that the asymptotic value does not depend on the underlying distribution. So, the numerical results of Chapter 2 may be more general than indicated there. In Chapter 4, we find an approximation to the hypothesis testing problem which does not depend on the distribution of the variable X but only on the correlation coefficient ((rho)) between X and Y. The approximation is thus found when we formulate the analogous problem in continuous time.