M-statistics

Chapter 6Multidimensional parameter

This chapter aims to generalize M-statistics to multidimensional parameter . First, we start with exact tests given the observed statistic value provided the density function is known up to a finite number of unknown parameters. Two versions are developed: the density level (DL) test and the unbiased test, as generalizations of the respective tests for the one-dimensional parameter from Section 2.3 and Section 3.1, respectively. The results related to the Neyman-Pearson lemma presented in Section 1.4 are used to prove the optimality of the DL test. Second, exact confidence regions, dual to the respective tests, are offered. Similarly to the single-parameter case, the point estimator is derived as a limit point of the confidence region when the confidence level approaches zero. A connection between maximum likelihood (ML) and mode estimators (MO) is highlighted. Four examples illustrate our exact statistical inference: simultaneous testing and confidence region for the mean and standard deviation of the normal distribution, two shape parameters of the beta distribution, two-sample binomial problem, and nonlinear regression. The last problem illustrates how an approximate profile statistical inference can be applied to a single parameter of interest.

6.1 Density level test

We aim to generalize an exact test from Section 2.3 with type I error ...