

Title of Thesis
Bayesian Approach of Testing Many Hypotheses
Concerning Parameters of Multivariate Normal Distribution 
Author(s)
Muntazim Abbas
Hashmi 
Institute/University/Department
Details Abdus Salam School of Mathematical Sciences / GC
University, Lahore 
Session 2011 
Subject Mathematics 
Number of Pages 127 
Keywords (Extracted from title, table of contents and
abstract of thesis) Optimality, Hypothesis, Normal,
Combination, Existence, Bayesian, Distribution, Multivariate,
Testing , Approach, Normal, Parameters, Concerning 
Abstract Multiple hypothesis
testing is an important topic in statistics.Therefore, the problem
addressed in this thesis is an important one.The Bayesian methods of
hypotheses testing are widely used for solving different problems,
and this technique is rather well developed. A lot of scientific
works are dedicated to the development of this method.Many
interesting and important results have been obtained in this field
by different authors.Despite of this fact there still remain a lot
of unsolved problems.For filling these gaps, in this thesis we
consider different problems of testing many hypotheses by the
Bayesian approach. In particular, in the Bayesian problem of many
hypotheses testing concerning all the parameters of multidimensional
normal distribution at correlation of observation results we have
obtained the following new results: the problem of computation of
the risk function were considered; the formulae
or calculation of multidimensional probability integrals by series
using the reduction of dimensionality to one without information
loss were derived; the formulae for calculation of product moments
for normalized normally distributed random values were derived; the
problems of existence and continuity of the probability distribution
law of linear combination of exponents of quadratic forms of the
normally distributed random vector, and, also, the problem of
finding the closed form of this law were considered; the existence
of this law and the opportunity of its unambiguous determination by
calculated moments of the appropriate random variable were proved;
the approximation of optimal regions of acceptance of hypotheses,
which significantly simplify the algorithms of realization of
general solutions of the task, is offered; the properties and
interrelations of the developed methods and algorithms were
investigated; the problem of choosing the loss function in the
Bayesian problem of many hypotheses testing was considered; the
results of sensitivity analysis of the considered Bayesian problem
are given; the calculation results for concrete examples, which show
the validity of the obtained results are given.Especially must be
emphasized that new sequential method of testing many hypotheses
based on special properties of regions of acceptance of hypotheses
in the conditional Bayesian task of testing many hypotheses is
offered.The results of research of the properties of this method are
given.They show the consistency, simplicity and optimality of the
obtained results in the sense of the chosen criterion, which
consists in the upper restriction of the probability of the error of
one kind and the minimization of the probability of the error of the
second kind. The examples of testing of hypotheses for the case of
the sequential independent sample from the multidimensional normal
law of probability distribution with correlated components are
cited. They show the high quality of the offered methods 
