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

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