Multiple Hypotheses LAO Testing for Many Independent Objects

Abstract

The procedure of many hypotheses logarithmically asymptotically optimal (LAO) testing for a model consisting of three or more independent objects is analyzed. It is supposed that M probability distributions are known and each object follows one of them independently of others. The matrix of asymptotic interdependencies (reliability-reliability functions) of all possible pairs of the error probability exponents (reliabilities) in optimal testing for this model is studied. This problem was introduced (and solved for the case of two objects and two given probability distributions) by Ahlswede and Haroutunian; it is a generalization of two hypotheses LAO testing problem for one object investigated

Multiple Objects: Error Exponents in Hypotheses Testing and Identification

“…The problem of hypotheses testing for the model consisting of two independent and of two strictly dependent objects (when they cannot admit the same distribution) with two possible hypothetical distributions were solved in [6]. In [39] the specific characteristics of the model consisting of K(≥ 2) objects each independently of others following one of given M (≥ 2) probability distributions were explored. In [47] the model composed by stochastically related objects was investigated.…”

“…Shannon information theory and mathematical statistics interaction revealed to be effective. This interplay is mutually fruitful, in some works results of probability theory and statistics were obtained with application of informationtheoretical methods and there are studies where statistical results provide ground for new findings in information theory [12], [14], [16]- [19], [35], [39], [49], [54], [57]- [59]. This paper can serve an illustration of application of information-theoretical methods in statistics: on one hand this is analogy in problem formulation and on the other hand this is employment of technical tools of proof, specifically of the method of types [15], [17].…”

A Neyman-Pearson Proper Way to Universal Testing of Multiple Hypotheses Formed by Groups of Distributions

“…Neyman-Pearson criterion of multiple hypotheses testing for discrete random variables was explored in [25]. In publications [1], [24] and [26], many hypotheses logarithmically asymptotically optimal (LAO) testing for the models consisting of many independent objects was investigated. Following Birgé [3], we called the sequence of tests logarithmically asymptotically optimal (LAO), when for given values of some reliabilities (error probability exponents) the test ensures the best values for the rest of them.…”

“…Hypotheses testing with no-match decision was considered by Gutman [16]. In papers [28]- [30] the results of researches of characteristics of LAO hypotheses testing with possibility of rejection of decision for some models with one or multiple objects, with side information are presented.…”

Optimal testing of statistical hypotheses and multiple familywise error rates

“…Ahlswede and Haroutunian [1,12,13] formulated some problems of multiple hypotheses testing and identification for many objects. Multiple hypotheses LAO testing for many independent objects is also investigated [14] and the multistage tests of multiple hypotheses are studied by Bartroff and Lai [2].…”