M-statistics

Chapter 5M-statistics for major statistical parameters

This chapter displays M-statistics in action: the maximum concentration (MC) and mode (MO) exact statistical inference are applied to major statistical parameters. Here M-statistics operates with CIs, tests, and point estimators under one methodological umbrella. We develop new exact optimal double-sided CIs and statistical tests and provide numerical algorithms to find the solutions with R codes found on GitHub. When the closed-form for the cdf does not exist we express the cdf via integral over the density and use numerical integration (integrate) to facilitate derivation with respect to the parameter. Real-life examples illustrate the application and how to call the respective R functions. An important aspect of our discussion is the derivation of the power function of the test, which is valuable from a theoretical perspective to compare the tests, and from an application point of view to the sample size determination given the power value. We do not provide a comprehensive literature review on statistical inference of parameters considered in this chapter but illustrate M-statistics.

5.1 Exact statistical inference for standard deviation

As was mentioned in the motivating example, the optimal CIs for normal variance were developed many years ago. Remarkably, the exact statistical inference for normal standard deviation is still underdeveloped. For example, the widespread CI for as the square root of the equal-tail ...