ABSTRACT
Probability theory has several fundamental branches that have been dealt with extensively in the literature associated with martingale techniques. Chapter 4 focuses on the following aspects that have straightforward connections with biostatistical theory and its applications: 1) A martingale can represent a statistic, say Mn , such that the expectation of a future outcome M n+1, which is calculated given knowledge regarding the present statistic Mn , is equal to the observed value Mn . For example, if a test statistic, say Tn , based on n observations satisfies this martingale property, under the null hypothesis H 0, then this can reduce the chances that Tk, k > n, will reject H 0 when more data points are available and H 0 is true. 2) The martingale machinery can extend different concepts based on sums of iid summands. 3) Martingale methodology plays a crucial role in the development and examination of various sequential procedures .
Chapter 4 outlines the following topics: Definitions and examples of martingale related components; The Optional Stopping Theorem and its corollaries; Applications of martingale theory for developing the principles of efficient retrospective and sequential procedures, including adaptive estimators and change point detection policies; A novel discovery in martingale transformations of testing strategies and martingale based comparisons between decision-making procedures.
