Asymptotic relatively more efficient test with auxiliary information: the case of the $Z$-test and the chi-square test

TL;DR

This paper explores how incorporating auxiliary information can asymptotically enhance the efficiency of the Z-test and chi-square test, demonstrating improvements through theoretical analysis and examples.

Contribution

It introduces a general framework for integrating auxiliary information into these tests, resulting in more efficient procedures in the sense of Pitman's ARE.

Findings

Efficiency of modified tests is improved asymptotically.

Theoretical analysis confirms the advantage of auxiliary information.

Examples illustrate practical application of the method.

Abstract

The main goal of this article is to study how an auxiliary information can be used to improve the efficiency of two famous statistical tests: the $ Z$-test and the chi-square test. Many definitions of auxiliary information can be found in the statistical literature. In this article, the notion of auxiliary information is discussed from a very general point of view and depends on the relevant test. These two statistical tests are modified so that this information is taken into account. It is shown in particular that the efficiency of these new tests is improved in the sense of Pitman's ARE. Some statistical examples illustrate the use of this method.