New sequential methods of multiple testing problems based on special properties of hypotheses acceptance regions in the constrained Bayesian tasks of testing hypotheses are offered. Results of an investigation on the properties of one of these methods are given. They show the consistency, simplicity, and optimality of the results obtained in the sense of the chosen criterion. The essence of the criterion is to restrict from above the probability of the error of one type and to minimize the probability of the error of the second type. The facts of the validity of the suitable properties of the method are proved. Examples of testing of hypotheses for the sequentially obtained independent samples from the multivariate normal distribution with correlated components are cited. They show the high quality of the proffered methods. The results of the Wald sequential method are given for the examples with two hypotheses and compared with the results obtained by the proffered method.

The Methods of Sequential Analysis of Bayesian Type for the Multiple Testing Problem

A sequential method based on constrained Bayesian methods is developed for testing multiple hypotheses. It controls the family-wise error rate and the family-wise power in a more accurate form than the Bonferroni or intersection scheme using the ideas of step-up and step-down methods for multiple comparisons of sequential designs. The new method surpasses the existing testing methods proposed earlier in a substantial reduction of the expected sample size.

Constrained Bayesian Method for Testing Multiple Hypotheses in Sequential Experiments

The paper deals with the problem of electronic verification of people on the basis of measurement information of a fingerprint reader and new approaches to its solution. The offered method guaranties the restriction of error probabilities of both type at the desired level at making a decision about permitting or rejecting the request on service in the system. On the basis of investigation of real data obtained in the real biometrical system, the choice of distribution laws is substantiated and the proper estimations of their parameters are obtained. Using chosen distribution laws, the normal distribution for measurement results of characteristics of the people having access to the system and the beta distribution for the people having no such access, the optimal rule based on the Constrained Bayesian Method (CBM) of making a decision about giving a permission of access to the users of the system is justified. The CBM, the Neyman–Pearson and classical Bayes methods are investigated and their good a...

Verification in biometric systems: problems and modern methods of their solution

The program packages of realization of mathematical models of pollutants transport in riversand for identification of river water excessive pollution sources located between two controlled cross sections of the river will be considered and demonstrated The software has been developed by the authors on the basis of mathematical models of pollutant transport in the rivers and statistical hypotheses test shy ing methods The identification al shy go shy rithms were elaborated with the supposition that the pollution sources discharge dif shy fe shy rent compositions of pollutants or at the identical com shy po shy sition different propor shy tions of pollutants into the rivers One two and three dimensional advection diffusion mathematical models of river water quality formation both under classical and new original boundary conditions are realized in the package New finite difference schemes of calculation have been developed and the known ones have been improved for these mathematical models At the same time a number of important problems which provide practical realization high accuracy and short time of obtaining the solution by computer have been solved Classical and new constrained Bayesian methods of hypotheses testing for identification of river water excessive pollution sources are realized in the appropriate software The packages are designed as a up to date convenient reliable tools for specialists of various areas of knowledge such as ecology hydrology building agriculture biology ichthyology and so on They allow us to calculate pollutant concentrations at any point of the river depending on the quan shy shy tity and the conditions of discharging from several pollution sources and to identify river water excessive pollution sources when such necessity arise

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Software for pollutants transport in rivers and for identification of excessive [ollution sources

The Comparison analysis of the Wald’s and Bayestype sequential methods for testing hypotheses is o ffered. The merits of the new sequential test are: universality which onsists in optimality (with given criteria) and uniformity of decision-making regions for any number of hypotheses; simplicity, c onvenience and uniformity of the algorithms of their realization; reliability of the obtained results and an opportunity of providing th e errors probabilities of desirable values. There are given the Computation results of concrete examples which confirm the abov e-stated characteristics of the new method and characterize the considered methods in regard to each other. Keywords—Errors of types I and II, likelihood ratio, the Bay es Type Sequential test, the Wald’s sequential test, a veraged number of observations.

Comparison Analysis of the Wald-s and the Bayes Type Sequential Methods for Testing Hypotheses

Specific features of the regions of acceptance of hypotheses in conditional Bayesian problems of statistical hypotheses testing are discussed. It is shown that the classical Bayesian statement of the problem of statistical hypotheses testing in the form of an unconditional optimizing problem is a special case of conditional Bayesian problems of hypotheses testing set in the form of conditional optimizing problems. It is also shown that, at acceptance of hypotheses in conditional problems of hypotheses testing, the situation is similar to the sequential analysis. It is possible an occurrence of the situation when the acceptance of a hypothesis with specified validity on the basis of the available information is impossible. In such a situation, the actions are similar to the sequential analysis, i.e. it is necessary to obtain additional information in the form of new observation results or to change the significance level of a test.

G. K. Kachiashvili

Papers

Specific features of regions of acceptance of hypotheses in conditional Bayesian problems of statistical hypotheses testing