127 research outputs found
Final solution to the problem of relating a true copula to an imprecise copula
In this paper we solve in the negative the problem proposed in this journal
(I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and
Systems, 278 (2015), 48-66) whether an order interval defined by an imprecise
copula contains a copula. Namely, if is a nonempty set of
copulas, then and are quasi-copulas and the pair
is an imprecise copula according to the
definition introduced in the cited paper, following the ideas of -boxes. We
show that there is an imprecise copula in this sense such that there is
no copula whatsoever satisfying . So, it is
questionable whether the proposed definition of the imprecise copula is in
accordance with the intentions of the initiators. Our methods may be of
independent interest: We upgrade the ideas of Dibala et al. (Defects and
transformations of quasi-copulas, Kybernetika, 52 (2016), 848-865) where
possibly negative volumes of quasi-copulas as defects from being copulas were
studied.Comment: 20 pages; added Conclusion, added some clarifications in proofs,
added some explanations at the beginning of each section, corrected typos,
results remain the sam
More on restricted canonical correlations
AbstractThe problem of the first canonical correlation between two random vectors subject to some natural constraints is treated in the paper. The problem is usually referred to as restricted canonical correlation. A new approach to solving the problem is given by translating it into a generalized eigenvalue problem with an n×n real symmetric matrix A and a positive definite matrix B of the same size
The solution of the Loewy-Radwan conjecture
A seminal result of Gerstenhaber gives the maximal dimension of a linear
space of nilpotent matrices. It also exhibits the structure of this space where
the maximal dimension is attained. Extensions of this result in the direction
of linear spaces of matrices with a bounded number of eigenvalues have been
studied. In this paper, we answer perhaps the most general problem of the kind
as proposed by Loewy and Radwan by solving their conjecture in the positive. We
give the dimension of a maximal linear space of matrices with no
more than eigenvalues. We also exhibit the structure of the space where
this dimension is attained
A full scale Sklar's theorem in the imprecise setting
In this paper we present a surprisingly general extension of the main result
of a paper that appeared in this journal: I. Montes et al., Sklar's theorem in
an imprecise setting, Fuzzy Sets and Systems, 278 (2015), 48--66. The main
tools we develop in order to do so are: (1) a theory on quasi-distributions
based on an idea presented in a paper by R. Nelsen with collaborators; (2)
starting from what is called (bivariate) -box in the above mentioned paper
we propose some new techniques based on what we call restricted (bivariate)
-box; and (3) a substantial extension of a theory on coherent imprecise
copulas developed by M. Omladi\v{c} and N. Stopar in a previous paper in order
to handle coherence of restricted (bivariate) -boxes. A side result of ours
of possibly even greater importance is the following: Every bivariate
distribution whether obtained on a usual -additive probability space or
on an additive space can be obtained as a copula of its margins meaning that
its possible extraordinariness depends solely on its margins. This might
indicate that copulas are a stronger probability concept than probability
itself.Comment: 16 pages, minor change
Coarse version of the Banach-Stone theorem
We show that if there exists a Lipschitz homeomorphism between the nets
in the Banach spaces and of continuous real valued functions on
compact spaces and , then the spaces and are homeomorphic
provided . By and we denote the
Lipschitz constants of the maps and . This improves the classical
result of Jarosz and the recent result of Dutrieux and Kalton where the
constant obtained is 17/16. We also estimate the distance of the map from
the isometry of the spaces and
Constructing copulas from shock models with imprecise distributions
The omnipotence of copulas when modeling dependence given marg\-inal
distributions in a multivariate stochastic situation is assured by the Sklar's
theorem. Montes et al.\ (2015) suggest the notion of what they call an
\emph{imprecise copula} that brings some of its power in bivariate case to the
imprecise setting. When there is imprecision about the marginals, one can model
the available information by means of -boxes, that are pairs of ordered
distribution functions. By analogy they introduce pairs of bivariate functions
satisfying certain conditions. In this paper we introduce the imprecise
versions of some classes of copulas emerging from shock models that are
important in applications. The so obtained pairs of functions are not only
imprecise copulas but satisfy an even stronger condition. The fact that this
condition really is stronger is shown in Omladi\v{c} and Stopar (2019) thus
raising the importance of our results. The main technical difficulty in
developing our imprecise copulas lies in introducing an appropriate stochastic
order on these bivariate objects
The linear preservers of real diagonalizable matrices
Using a recent result of Bogdanov and Guterman on the linear preservers of
pairs of simultaneously diagonalizable matrices, we determine all the
automorphisms of the vector space M_n(R) which stabilize the set of
diagonalizable matrices. To do so, we investigate the structure of linear
subspaces of diagonalizable matrices of M_n(R) with maximal dimension.Comment: 14 page
On approximate commutativity of spaces of matrices
The maximal dimension of commutative subspaces of is known.
So is the structure of such a subspace when the maximal dimension is achieved.
We consider extensions of these results and ask the following natural
questions: If is a subspace of and is an integer less
than , such that for every pair and of members of , the rank of
the commutator is at most , then how large can the dimension of
be? If this maximum is achieved, can we determine the structure of ? We
answer the first question. We also propose a conjecture on the second question
which implies, in particular, that such a subspace has to be an algebra,
just as in the known case of . We prove the proposed structure of if
it is already assumed to be an algebra.Comment: 17 page
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