Exponential increase of test power for Z-test and Chi-square test with auxiliary information
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The main goal of this article is to study how an auxiliary information can be used to improve the power of two famous statistical tests: the Z-test and the chi-square test. This information can be of any nature - probability of sets of partitions, expectation of a function, ... - and is not even required to be an exact information, it can be given by an estimate based on a larger sample for example. Some definitions of auxiliary information can be found in the statistical literature and will be recalled. In this article, the notion of auxiliary information is discussed here from a very general point of view. These two statistical tests are modified so that the auxiliary information is taken into account. One show in particular that the power of these tests is increased exponentially. Some statistical examples are treated to show the concreteness of this method.
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