Convex combinations of nilpotent triangular norms

In this paper we deal with the open problem of convex combinations of continuous triangular norms stated by Alsina, Frank, and Schweizer [C. Alsina, M.J. Frank, B. Schweizer, Problems on associative functions, Aequationes Math. 66 (2003) 128-140, Problems 5 and 6]. They pose a question whether a non-trivial convex combination of triangular norms can ever be a triangular norm. The main result of this paper gives a negative answer to the question for any pair of continuous Archimedean triangular norms with different supports. With the help of this result we show that a non-trivial convex combination of nilpotent t-norms is never a t-norm. The main result also gives an alternative proof to the result presented by Ouyang and Fang [Y. Ouyang, J. Fang, Some observations about the convex combination of continuous triangular norms, Nonlinear Anal., 68 (11) (2008) 3382-3387, Theorem 3.1]. In proof of the main theorem we utilize the Reidmeister condition known from the web geometry.     

Convex linear combinations of sequences of monic orthogonal polynomials

For a sequence of monic orthogonal polynomials (SMOP), with respect to a positive measure supported on the unit circle, we obtain necessary and sufficient conditions on a SMOP in order that a convex linear combination with be a SMOP with respect to a positive measure supported on the unit circle.

     

Convex combinations of long memory estimates from different sampling rates

This paper investigates convex combinations of long memory estimates from both the original and sub-sampled data. Sub-sampling is carried out by decreasing the sampling rate, which leaves the long memory parameter unchanged. Any convex combination of these sub-sample estimates requires a preliminary correction for the bias observed at lower sampling rates, reported by Souza and Smith (2002). Through Monte Carlo simulations, we investigate the bias and the standard deviation of the combined estimates, as well as the root mean squared error (RMSE). Combining estimates can significantly lower the RMSE of a standard estimator (by about 30% on average for ARFIMA (0, d, 0) series), at the cost of inducing some bias.     

Convex combinations of continuous t-norms with the same diagonal function

The problem of whether a non-trivial convex combination of two continuous t-norms with the same diagonal function can be a t-norm is studied. It is shown that in both cases–of two nilpotent and of two strict t-norms–a non-trivial convex combination of t-norms with common diagonal function is associative only if the two t-norms involved coincide. For general continuous t-norms a similar result follows. An example of a convex class of non-continuous t-norms is also included.     

On convergence for sequences of pairwise negatively quadrant dependent random variables

In this paper, some new results on complete convergence and complete moment convergence for sequences of pairwise negatively quadrant dependent random variables are presented. These results improve the corresponding theorems of S.X.Gan, P.Y.Chen (2008) and H.Y. Liang, C. Su (1999).     

Conditional limit theorems for conditionally linearly negative quadrant dependent random variables

From the ordinary notion of linearly negative quadrant dependence for a sequence of random variables, a new concept called conditionally linearly negative quadrant dependence is introduced. The relation between the two kinds of dependence is answered by examples, that is, the linearly negative quadrant dependence does not imply the conditionally linearly negative quadrant dependence, and vice versa. The fundamental properties of conditionally linearly negative quadrant dependence are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional exponential inequalities, conditionally complete convergence results and a conditional central limit theorem stated in terms of conditional characteristic functions are established.     

Convergence properties for arrays of rowwise pairwise negatively quadrant dependent random variables

In this paper the authors study the convergence properties for arrays of rowwise pairwise negatively quadrant dependent random variables. The results extend and improve the corresponding theorems of T.C. Hu, R. L. Taylor: On the strong law for arrays and for the bootstrap mean and variance, Int. J. Math. Math. Sci 20 (1997), 375–382.     

Strong law of large numbers for pairwise positive quadrant dependent random variables

We give the rate of convergence in the strong law of large numbers for pairwise positive quadrant dependent random variables and contemporaneous functions of these variables. Several examples of applications are given.      

Shuffles of copulas

We show that every copula that is a shuffle of Min is a special push-forward of the doubly stochastic measure induced by the copula M. This fact allows to generalize the notion of shuffle by replacing the measure induced by M with an arbitrary doubly stochastic measure, and, hence, the copula M by any copula C.     

Approximation of bivariate copulas by patched bivariate Fréchet copulas

Bivariate Fréchet (BF) copulas characterize dependence as a mixture of three simple structures: comonotonicity, independence and countermonotonicity. They are easily interpretable but have limitations when used as approximations to general dependence structures. To improve the approximation property of the BF copulas and keep the advantage of easy interpretation, we develop a new copula approximation scheme by using BF copulas locally and patching the local pieces together. Error bounds and a probabilistic interpretation of this approximation scheme are developed. The new approximation scheme is compared with several existing copula approximations, including shuffle of min, checkmin, checkerboard and Bernstein approximations and exhibits better performance, especially in characterizing the local dependence. The utility of the new approximation scheme in insurance and finance is illustrated in the computation of the rainbow option prices and stop-loss premiums.     

Spatially homogeneous copulas

We consider spatially homogeneous copulas, i.e. copulas whose corresponding measure is invariant under a special transformations of \([0,1]^2\), and we study their main properties with a view to possible use in stochastic models. Specifically, we express any spatially homogeneous copula in terms of a probability measure on [0, 1) via the Markov kernel representation. Moreover, we prove some symmetry properties and demonstrate how spatially homogeneous copulas can be used in order to construct copulas with surprisingly singular properties. Finally, a generalization of spatially homogeneous copulas to the so-called (m, n)-spatially homogeneous copulas is studied and a characterization of this new family of copulas in terms of the Markov \(*\)-product is established.

     

Some mean convergence and complete convergence theorems for sequences of m-linearly negative quadrant dependent random variables

The structure of linearly negative quadrant dependent random variables is extended by introducing the structure of m-linearly negative quadrant dependent random variables (m = 1, 2, …). For a sequence of m-linearly negative quadrant dependent random variables {X _{n, n} ? 1} and 1 < p < 2 (resp. 1 ? p < 2), conditions are provided under which $n^{ - 1/p} \sum\limits_{k = 1}^n {\left( {\left. {X_k - } \right|EX_k } \right) \to } 0$ in L ¹. Moreover, for 1 ? p < 2, conditions are provided under which $n^{ - 1/p} \sum\limits_{k = 1}^n {\left( {X_k - EX_k } \right)}$ converges completely to 0. The current work extends some results of Pyke and Root (1968) and it extends and improves some results of Wu, Wang, and Wu (2006). An open problem is posed.     

Goodness-of-fit tests for copulas

This paper defines two distribution free goodness-of-fit test statistics for copulas. It states their asymptotic distributions under some composite parametric assumptions in an independent identically distributed framework. A short simulation study is provided to assess their power performances.     

Are copulas unimodal?

Three types of unimodality (central convex, block, and star) are considered and the corresponding sets of unimodal copulas determined. Examples of star unimodal copulas, absolutely continuous, with a nonnull singular part, and even singular, are given. Necessary and sufficient conditions for a diagonal to be the diagonal section of a star unimodal copula are also indicated. Attention is also paid to the Archimedean case.     

Construction of asymmetric multivariate copulas

In this paper we introduce two methods for the construction of asymmetric multivariate copulas. The first is connected with products of copulas. The second approach generalises the Archimedean copulas. The resulting copulas are asymmetric and may have more than two parameters in contrast to most of the parametric families of copulas described in the literature. We study the properties of the proposed families of copulas such as the dependence of two components (Kendall’s tau, tail dependence), marginal distributions and the generation of random variates.     

Tails of multivariate Archimedean copulas

A complete and user-friendly directory of tails of Archimedean copulas is presented which can be used in the selection and construction of appropriate models with desired properties. The results are synthesized in the form of a decision tree: Given the values of some readily computable characteristics of the Archimedean generator, the upper and lower tails of the copula are classified into one of three classes each, one corresponding to asymptotic dependence and the other two to asymptotic independence. For a long list of single-parameter families, the relevant tail quantities are computed so that the corresponding classes in the decision tree can easily be determined. In addition, new models with tailor-made upper and lower tails can be constructed via a number of transformation methods. The frequently occurring category of asymptotic independence turns out to conceal a surprisingly rich variety of tail dependence structures.     

A class of multivariate copulas with bivariate Fréchet marginal copulas

In this paper, we present a class of multivariate copulas whose two-dimensional marginals belong to the family of bivariate Fréchet copulas. The coordinates of a random vector distributed as one of these copulas are conditionally independent. We prove that these multivariate copulas are uniquely determined by their two-dimensional marginal copulas. Some other properties for these multivariate copulas are discussed as well. Two applications of these copulas in actuarial science are given.     

A structure theorem on bivariate positive quadrant dependent distributions and tests for independence in two-way contingency tables

In this paper, the set of all bivariate positive quadrant dependent distributions with fixed marginals is shown to be compact and convex. Extreme points of this convex set are enumerated in some specific examples. Applications are given in testing the hypothesis of independence against strict positive quadrant dependence in the context of ordinal contingency tables. The performance of two tests, one of which is based on eigenvalues of a random matrix, is compared. Various procedures based upon certain functions of the eigenvalues of a random matrix are also proposed for testing for independence in a two-way contingency table when the marginals are random.