John Bather

Sequential procedures for comparing several medical treatments

The use of sequential methods in clinical trials allows inferior treatments to be eliminated early. From an ethical point of view, the advantages are substantial. However, early stopping induces estimation bias and a deterioration in precision because of reduced sample sizes. This paper considers the problem of determining which of k ≥ 2 treatments with Bernoulli responses has the highest probability of success. Two sequential procedures are investigated and compared with a fixed—sample procedure. Various properties are derived and illustrated for the cases k =2,3 and 5. It is shown that the sequential procedures can achieve a pattern of error probabilities equivalent to the fixed—sample procedure for a much lower level of expected successes lost. Approximations for the bias and standard deviation of estimators of treatment differences are obtained by using results about the distribution of stopping times for a normal process.

OPTIMAL STOPPING OF BROWNIAN MOTION: A COMPARISON TECHNIQUE

Publisher Summary This chapter discusses the general theory of optimal stopping for Brownian motion. It explains why many sequential decision problems in discrete time lead to stopping problem when they are formulated in continuous time. The solution of the limiting form of the problem can be constructed, in principle, by taking the union of certain open sets and this approach is closely related to the comparison technique. The chapter describes several Markov processes, each of which is capable of producing a sequence of rewards, and the problem of choosing just one of them at each stage so as to maximize the total discounted expectation. Each separate process has an index, depending on its state, and the optimal allocation policy is then defined by choosing the process with highest index at every stage. There is a strong connection between multi-armed bandits and stopping problems because the index for a single Markov process must be determined by solving a family of stopping problems.

The effect of truncation on a sequential test for the drift of brownian motion

Lerche (1986) investigated a sequential testing problem concerned with deciding the sign of a normal mean. He demonstrated that a parabolic boundary which corresponds to

Stoppong rules and observed significance levels

Abstract : It is well known how to combine the significance levels observed in a number of independent experiments. When this number is a random variable determined by a stopping rule, the observed significance level can still be calculated if there is an acceptable ordering of the points in the extended sample space. But what can be said if the stopping time is ill-defined? This paper obtains explicit lower bounds on the level of significance by considering orderings based on a family of alternative hypotheses. These bounds give some measure of the effect of failing to specify the stopping rule in advance. Keywords: Stochastic processes.

Poission search models and the evaluation of stopping rules

Mathematically convenient models for a search can be based on a prior representation of the hidden objects and their values by a mixture of Poisson processes. In simple cases, we can find the corresponding Bayes procedures to maximise expected net gains, allowing for the cost of searching. More generally, optimal stopping rules for the search are difficult to construct and evaluate. We also investigate a procedure based on the asymptotic behaviour of the system when the number of hidden objects is large. This is shown to provide a reasonably effective stopping rule over a wide range of conditions.