Separable Statistics in a Polynomial Scheme. I

Abstract

In statistical analysis of discrete sequences, the problem often arises of testing the hypothesis H0 that the given sequence is obtained from a scheme of independent trials with N outcomes whose probabilities are q (qa, qz, ", qr) against the competing composite hypothesis H ={H(p)}, in which it is still assumed that the trials are independent and homogeneous, but where the probabilities p (pa, p, ., pu) of the corresponding trials are left arbitrary. A rather well developed theory is available for the case where the number of possible outcomes N is finite or whose growth, as the number of trials n increases, is such that the quantity a min np,, also increases without limit.<m Practical requirements of recent years have posed the problem of investigating schemes with a large number of outcomes. In this situation, reduction of the problem to the classical case by combining outcomes into groups does not always turn out to be appropriate, because of the loss of information about the scheme, and also because of the introduction of a certain amount of subjectivity in the way the outcomes are combined. A characteristic feature of this case is the fact that although the number of trials n is indeed large, it nevertheless does not always guarantee that the quality a increases without limit. Under these nontraditional conditions, the classical methods of solution, based on the asymptotic normality of the vector u (u, t,2,'' ", ur) of the number of realizations of the outcomes, are no longer efficacious, and new approaches to the solution of the problem are needed. Here great interest attaches both to the cases where all the mean values np,, are bounded and to the cases where these quantities approach 0 as n increases. We can refer to the schemes where the quantities np,, are bounded as small sample schemes (see [5]). These schemes have been studied comparatively little as compared with classical schemes, although a rather deep investigation of a whole series of statistics involving the frequences has been made under these conditions. Here we mention only a few papers (see [1]-[14]) related to the statistics in question, referring the reader to the more detailed bibliography given in [9]. DEFINITION. A function (statistic) L=L(u)=L(ua, uz,'", UN)

Generalized Error Exponents for Small Sample Universal Hypothesis Testing

“…An important class of tests is based on separable statistics [37]: this is a test statistic of the form (9) where (10) is the empirical distribution, and is any function that does not depend on except via its argument. General theorems on asymptotic distributions and asymptotic moments of separable statistics are available in [37].…”

“…General theorems on asymptotic distributions and asymptotic moments of separable statistics are available in [37]. Large deviations analysis for the case is given in [32] and [33].…”

“…Since is analytic in , the moment generating function of can be obtained via Cauchy's theorem: (37) where the integration is carried out along any closed contour around in the complex plane. These arguments lead to the following lemma.…”

On calculation of exact distributions of decomposable statistics in the multinomial scheme

“…where s m is the corresponding sum (12), is satisfied, then according to the central limit theorem for the U -statistics [ (15) is satisfied for all m D 2; 3; : : : if the alternative H q does not belong to the set of alternatives H which is defined in [11] and is constructed as follows. We take different subsets A f1; 2; : : : ; N g, 2 Ä jAj < N , where jAj is the number of elements in A; for i D 1; : : : ; N we construct the vector q D .q 1 ; : : : ; q N / with the components and include the alternative H q corresponding to the vector q in H .…”

Weak Convergence Analysis of Asymptotically Optimal Hypothesis Tests

“…The other setting where n = o(k) and k = o(n 2 ) (the latter is required for any test to be consistent) has been used to model the case where the number of samples is significantly small compared to the size of the alphabet. See [12], [28], [29] for earlier weak-convergence type of results. Many recent works from the information theory community and theoretical computer science community [4]- [6], [13], [30]- [34] focus on asymptotic theoretical achievability and converse results, as well as new test designs.…”