Copulas
for which
are called
copulas, of which well-studied examples are idempotent copulas and complete dependence copulas. As such, we shall work mainly with the topology induced by the modified ...Sobolev norm, with respect to which the class
of pre-idempotent copulas is closed and the class of factorizable copulas is a dense subset of
. Identifying copulas with Markov operators on
, the one-to-one correspondence between pre-idempotent copulas and partial isometries is one of our main tools. In the same spirit as Darsow and Olsen’s work on idempotent copulas, we obtain an explicit characterization of pre-idempotent copulas, which is split into cases according to the atomicity of its associated
-algebras, where the nonatomic case gives all factorizable copulas and the totally atomic case yields conjugates of ordinal sums of copies of the product copula.
Two (continuous) random variables X and Y are implicitly dependent if there exist Borel functions α and β such that α∘X=β∘Y almost surely. The copulas of such random variables are exactly the copulas ...that are factorizable as the ∗-product of a left invertible copula and a right invertible copula. Consequently, every implicit dependence copula assigns full mass to the graph of f(x)=g(y) for some measure-preserving functions f and g but the converse is not true in general.
We obtain characterizations of a copula C assigning full mass to the graph of f(x)=g(y) in terms of a partial factorizability of its Markov operator TC and in terms of the non-atomicity of two newly defined associated σ-algebras σC and σC∗, in which case C is called non-atomic. As an application, we give a broad sufficient condition under which a copula with fractal support has an implicit dependence support. Under certain extra conditions, we explicitly compute the left invertible and right invertible factors of the copula with fractal support.
•Non-atomic bivariate copulas are defined in terms of two associated σ algebras.•They are characterized in terms of its mass concentration and in terms of a partial factorizability of its Markov operators.•Sufficient conditions for copulas with fractal support to be non-atomic and to be factorizable are given.
Based on the Zermelo-Fraenkel system of axioms ZF, we introduce a theory of essential closures. It is a generalization of the concept of topological closures in which a set may not be contained in ...its essential closure. A typical essential closure collects all points that are essential with respect to a submeasure; hence it is called a submeasure closure. One of our main results states that a "nice" essential closure must be a submeasure closure. Many examples of known and new submeasure closures are discussed and their applications are demonstrated, especially in the study of the supports of measures.
A conditional variance is an indicator of the level of independence between two random variables.
We exploit this intuitive relationship and define a measure v which is almost a measure of mutual ...complete
dependence. Unsurprisingly, the measure attains its minimum value for many pairs of non-independent ran-
dom variables. Adjusting the measure so as to make it invariant under all Borel measurable injective trans-
formations, we obtain a copula-based measure of dependence v* satisfying A. Rényi’s postulates. Finally, we
observe that every nontrivial convex combination of v and v* is a measure of mutual complete dependence.
Factorizable non-atomic copulas Printechapat, Tanes; Sumetkijakan, Songkiat
Statistics & probability letters,
December 2018, 2018-12-00, Letnik:
143
Journal Article
Recenzirano
Generalizing the notion of invariant sets by Darsow and Olsen, Sumetkijakan studied a subclass of singular copulas, the so-called non-atomic copulas, defined via its associated σ-algebras. It was ...shown that the Markov operator of every non-atomic copula is partially factorizable, i.e. it is the composition of left and right invertible Markov operators on a subspace of L1(0,1) depending on the copula. Here, we further investigate the associated σ-algebras of the product of certain copulas and obtain (1) a sharper result on the partial factorizability of non-atomic copulas and (2) the existence and uniqueness of a completely factorizable copula that shares the same set of associated σ-algebras as that of a given non-atomic copula.
Extremality and factorizability of Markov operators Printechapat, Tanes; Santiwipanont, Tippawan; Sumetkijakan, Songkiat
Journal of mathematical analysis and applications,
11/2020, Letnik:
491, Številka:
2
Journal Article
Recenzirano
We prove that the extreme Markov operators that are factorizable as a composition of left and right invertible Markov operators must be one-sided invertible. It clearly follows that Markov operators, ...i.e. their corresponding doubly stochastic measures, with hairpin support are either invertible or non-factorizable. An alternative and more direct proof of this fact for Markov operators with full hairpin support is also given. We also demonstrate that there exists extreme, hence singular, Markov operator that is totally atomic.
The copula of two uniform-(0,1) random variables U and V is called implicit dependence if there exist measure-preserving transformations α and β on 0,1 such that α(U)=β(V) almost surely. ...Relationships between the implicit dependence copula C(U,V) and the complete dependence copulas Ce,α and Cβ,e, where e is the identity map, are investigated. In the case where both α and β are countably piecewise monotonic surjections, we obtain a characterization of implicit dependence copulas in terms of generalized Markov products of Ce,α and Cβ,e.
It was proved by Jang et al. that various chains of one-parameter distributions converge to Benford’s law. We study chains of truncated distributions and propose another approach, using a recent ...convergence result of the Lerch transcendent function, to proving that they converge to Benford’s law for initial Beta distributions with parameters
and 1.
A construction for providing single dyadic orthonormal wavelets in Euclidean space ℝd is given. It is called the general neighborhood mapping construction. The fact that one wavelet is sufficient to ...generate an orthonormal basis for L2(ℝd) is the critical issue. The validity of the construction is proved, and the construction is implemented computationally to provide a host of examples illustrating various geometrical properties of such wavelets in the spectral domain. Because of the inherent complexity of these single orthonormal wavelets, the method is applied to the construction of single dyadic tight frame wavelets, and these tight frame wavelets can be surprisingly simple in nature. The structure of the spectral domains of the wavelets arising from the general neighborhood mapping construction raises a basic geometrical question. There is also the question of whether or not the general neighborhood mapping construction gives rise to all single dyadic orthonormal wavelets. Results are proved giving partial answers to both of these questions.
Patched approximations and their convergence Chaidee, Nattakarn; Santiwipanont, Tippawan; Sumetkijakan, Songkiat
Communications in statistics. Theory and methods,
05/2016, Letnik:
45, Številka:
9
Journal Article
Recenzirano
Patched approximations of copulas unify ordinal sums, shuffles of Min, checkerboard, and checkmin approximations. We give a characterization of patched approximations and an error bound of the ...approximations in Sobolev norm. Patched approximations with uniform marginal conditional distributions are shown to arise naturally. We prove that these uniform patched approximations converge uniformly and in the Sobolev norm. The latter convergence is settled by showing the convergence almost everywhere of the first partial derivatives. We also show that the independence copula can be approximated by conditional mutual complete copulas in the Sobolev norm.
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