ABSTRACT
The purpose of this chapter is twofold with respect to: (1) Our desire to introduce a very powerful theoretical scheme for constructing statistical procedures, and (2) We would like to show arguments against statistical stereotypes. For example, one generally can state that if a statistical test has a reasonable fixed Type I error rate then the test cannot be of power one. In this chapter, a test with power one is presented. In Chapter 8 we apply the law of the iterated logarithm to judge statistical techniques. The material presented in this chapter may seem overly technical, however we encourage the reader to study the methodological approaches introduced in Chapter 8 in order to obtain beneficial skills in developing advanced statistical procedures. Chapter 8 considers the Ville and Wald inequality. This inequality is extended and rewritten in terms of sums of iid random variables. The statistical significance of the obtained results is introduced, showing the following procedures related to: confidence sequences and tests with uniformly small error probability for the mean of a normal distribution with known variance; and confidence sequences for the median.
