Abstract:
Multiple hypothesis testing is an important topic in statistics. Therefore, the
problem addressed in this thesis is an important one. It is also a topic in which
it is difficult to make a significant improvement, for various reasons. One reason is
that often different users may have different objectives and with multiple hypotheses
there is no unique objective function. In the thesis is recognized this fact and as the
objective functions, estimated the quality of made decisions, are used minimization
of the probabilities of the errors of one kind at restrictions of the probabilities of the
errors of second kind. Such approach is a new one which causes the uniqueness of
the regions of acceptance of hypotheses and, consequently, improves the quality of
hypothesis testing.
Thus conditional Bayesian tasks of testing many hypotheses are stated and solved.
The concept of conditionality is used for designation of the fact that the Bayesian
tasks are stated as conditional optimization problems where the probabilities of one-
type errors are restricted and, under such conditions, the probabilities of second-type
errors are minimized. The properties of obtained decision rules are investigated, and,
on their basis, it is shown that the classical Bayesian problem of hypotheses testing
is a special case of the considered. The calculation results of concrete examples have
shown that the qualities of offered conditional tasks surpass the quality of the classical
Bayesian task. They completely confirm the results of theoretical investigations. The
convenience, simplicity and naturalness of introduction of similar gradation Kiefer,
viiviii
(1977) by the level of certainty of hypotheses testing on the basis of concrete obser-
vation result are shown in offered conditional tasks.
Quasi-optimal procedures of many hypotheses testing are offered. They signif-
icantly simplify Bayesian algorithms of hypotheses testing and computation of the
risk function. The obtained general solutions are reduced to concrete formulae for
multivariate normal distribution of probabilities. The methods of approximate com-
putation of the risk functions in Bayesian tasks of testing many hypotheses are of-
fered. The properties and interrelations of the developed methods and algorithms
are investigated. On the basis of simulation, the validity of the obtained results and
conclusions made is shown.
The results of sensitivity analysis of the conditional Bayesian problems are given
and their advantages and drawbacks are considered.