On some statistical properties of the “Book Stack” transformation

Аннотация

This paper is devoted to studying the statistical properties of the “Book Stack” transformation proposed by B.Ya. Ryabko (Probl. Inf. Transm., 1980, vol. 16, no. 4) as a data compression method. The same transformation has been used by Ryabko and A.I. Pestunov (Probl. Inf. Transm., 2004, vol. 40, no. 1) to construct the similarly named statistical test. This test is designed for the verification of the null hypothesis that an available input i.i.d. sample corresponds to a discrete uniform distribution with a known support. They propose to verify this hypothesis for a new sample obtained via the Book Stack transformation instead of the input sample. This gives rise to the natural problem of comparing the results given by the same statistical test in the application to input and output samples. If the null hypothesis is true, these procedures prove to be equivalent; however, this is actually not the case anymore when there are some violations of this hypothesis. The results of comparing the criteria surely depend on the class of the alternatives considered. This paper deals with the natural alternative consisting of the fact that the initial replicated sample corresponds to a discrete, albeit, nonuniform, distribution with a fixed support. It has been demonstrated that some standard criteria for the verification of the null hypothesis prove to be more powerful for an input sample in comparison with a transformed sample. In particular, this takes place for the likelihood ratio criterion and (with some formal constraints) the χ²-criterion.