Testen statistischer Funktionale f\"ur Zweistichprobenprobleme

TL;DR

This paper develops a theoretical framework for testing statistical functionals in non-parametric two-sample problems, proposing tests with optimal properties and analyzing their asymptotic power and computability.

Contribution

It introduces new tests for differentiable functionals in two-sample problems, demonstrating their asymptotic optimality and applicability to distribution-free scenarios.

Findings

Proposed tests are locally asymptotically most powerful.

Derived asymptotic power functions for various alternatives.

Developed distribution-free asymptotic optimal tests for key functionals.

Abstract

The theory of testing statistical functionals is developed for non-parametric two-sample problems. For differentiable real-valued statistical functionals, some tests for the one-sided and two-sided cases are proposed and studied. The asymptotic power function is computed along implicit alternatives and hypotheses and compared with upper bounds for the power functions of tests for limit experiments. It is shown that the proposed tests are locally asymptotically most powerful under weak assumptions. For some important test problems, the distribution-free asymptotic optimal tests are developed. Among other things, the products and sums of von Mises functionals and the Wilcoxon functional are treated. We pay special attention to the computability of the tests.