الفهرس 
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Abstract
Multivariate extreme value theory is perhaps the only known toolbox for analyzing
several extremal events simultaneously. Generally, the ordered multivariate data
subject is an active eld of research in theoretical and applied statistics. The ordered
data may belong to the usual model of order statistics (see, Galambos, 1987 and David
and Nagaraja, 2003) or its extensions such as the model of generalized order statistics
(e.g. Kamps, 1995 and Burkschat et al., 2003). Moreover, the ordered data may arise
from a common distribution function (DF) or it may be dependent on non-identical
multivariate data (e.g. Barakat, 2009).
In this work, we are focus on the study of the model of multivariate order statistics.
The study will extended to distributional theory and the asymptotic distributional
theory. It is known that there is no any natural basis for ordering multivariate
data. Therefore, the rst obstacle that encounters the researchers in studying the
subject of ordered multivariate data is to extend the univariate order concepts to the
higher dimensional situation. Barnet (1976) presented a fourfold classi cation of subordering
principles for multivariate random vectors. These principles can be classi ed
as, Marginal Ordering (M-ordering), Reduced Ordering (R-ordering), Conditional Ordering
(C-ordering) and Partial Ordering (P-ordering).
In this study, we are concerned with ordered multivariate data based on R-ordering
principle. Speci cally, ordering in the norm sense.
This thesis consists of ve chapters:
Chapter 1: This is an introductory chapter, in which we give some de nitions and
theorems of norms and D-norms (Section 1.1). Then, we give some de nitions and theorems
of order statistics and record values which will be needed in our work (Section
1.2). In the last section of this chapter, an introduction about multivariate ordered
data is given, also we review the work of Barakat (2001) (concerning the asymptotic
distribution theory of bivariate order statistics) and the work of Bairamov and Gebizlioglu
(1997) (concerning the ordering of random vectors in a norm sense).
Chapter 2: In this chapter, we investigate the asymptotic behavior of the extremes
of multivariate data by using the R-ordering principle. When, the sup-norm is used,
we reveal the interrelation between the R-ordering and M-ordering principles. The
asymptotic behavior of the maximum sup-norms corresponding to the bivariate data
is completely determined.
Chapter 3: The asymptotic behavior of the intermediate and central order statistics
of bivariate data by using the R-ordering principle is investigated in this chapter.
When, the sup-norm is used, we reveal the interrelation between the R-ordering and
M-ordering principles. The asymptotic behavior of the intermediate and central bivariate
order statistics based on sup-norms is completely determined. Moreover, new
results concerning the univariate intermediate order statistics are given (Section 3.1).
Chapter 4: In this chapter, we investigate the asymptotic behavior of multivariate
record values by using the R-ordering principle. Necessary and sucient conditions
for the weak convergence of multivariate record values based on sup-norm are determined
and some illustrative examples are given.
Chapter 5: In this chapter, it is proved that the weak convergence of multivariate extremes
by using the sup-norm implies the convergence of those multivariate extremes
in an arbitrary D-norm to the same type-limits. As a consequence of this result,
the asymptotic behavior of the extremes of a multivariate data by using any logistic
norm is completely determined. Moreover, the same result for bivariate intermediate
and multivariate record values is proved. Finally, we give an application to real data
illustrates and corroborates the theoretical results of the thesis.