Jozef Komorník

Asymptotic Periodicity of Markov and Related Operators

The aim of this paper is to provide a unified exposition of some results in the theory of asymptotic behaviour of Markov (and related) operators. These results followed the invited address of A. Lasota at ICM ’82. He introduced the notion of constrictive operators saying that an operator P is weakly (strongly) constrictive if there exists a weakly (strongly) compact set F such that all trajectories of densities converge in L 1 norm to F. Lasota et al. investigated the asymptotic periodicity of Markov operators and proved that it holds for (i) strongly constrictive Markov operators [14] (ii) weakly constrictive Frobenius-Perron operators [19].

CLOSED SUBSPACES, POLYNOMIAL OPERATORS IN THE SHIFT, AND ARMA REPRESENTATIONS

This paper is concerned with the representation of system behaviors by equations involving polynomial shift operators. In particular, the question of the elimination of latent (i.e., auxiliary) variables from an ARMA representation is considered for the case of multidimensional systems.

Perturbation of bivariate copulas

New types of constructions of bivariate copulas are introduced, discussed and exemplified. Based on a given copula C, we look for its perturbation into another copula C H , possibly close to C. A special stress is put on the perturbation of the basic copulas M, W, ?. Our results generalize several methods known from the literature, such as the Farlie-Gumbel-Morgenstern copula family, for example. An illustrative example when fitting copulas to real data is also added.

On Some Construction Methods for Bivariate Copulas

We propose a rather general construction method for bivariate copulas, generalizing some construction methods known from the literature. In some special cases, the constraints ensuring the output of the proposed method to be a copula are given. Our approach opens several new problems in copula theory.

Dependence measures for perturbations of copulas

Abstract In this paper (which is a substantially extended version of a conference paper from AGOP 2015), we investigate the effects of specific class of perturbations of bivariate copulas on several measures of dependence (Spearmans rho, Blomqvists beta, Ginis gamma, Kendalls tau), and tail dependence along both diagonal sections. It is demonstrated that the influence of the perturbation parameter on the values of the first three of the above coefficients of dependence is linear, while on the last one it is quadratic. Interesting numerical analyses for several important classes of Archimedean copulas are presented. It is also demonstrated that the considered perturbations do not change the coefficients of tail dependencies along the main diagonal but linearly reduce their values along the second diagonal. An interesting possible application for analyzing dependencies along the second diagonal of copulas represent insurance data, where censoring introduces a negative dependence between the investigated components of the claims. As a by-product, we present a new class of perturbations of copulas that linearly reduce the more popular coefficients of tail dependencies along the main diagonal, while preserving their values along the second diagonal. Subsequently using suitable elements of both above classes of perturbations, any original copula can be transformed to a resulting one, having coefficients of tail dependencies along both diagonals linearly reduced (with any couple of preselected linear proportions from [ 0 , 1 ] ).

Families of Perturbation Copulas Generalizing the FGM Family and Their Relations to Dependence Measures

In this paper we provide an extension of special parametric class of perturbations of an arbitrary copula (given in [3]) that represent a partial generalization of the FGM family of copulas for parameters from the unit interval. However the FGM family is defined for parameters from the interval \([-1,1]\). We present a construction of perturbations of an arbitrary copula also for parameters from the interval \([-1,0]\) so that together with the former family of perturbations of copulas we get a generalization of the FGM family for the whole interval \([-1,1]\). We also investigated the influence of the parameters of the introduced class of perturbations of copulas on several measures of dependence (Spearman’s rho, Blomqvist’s beta, Gini’s gamma, Kendall’s tau).

Multidimensional Copula Models for Parallel Development of the Us Bond Market Indices

Abstract Stock and bond markets co-movements have been studied by many researchers. The object of our investigation is the development of three U.S. investment grade corporate bond indices. We concluded that the optimal 3D as well as partial pairwise 2D models are in the Student class with 2 degrees of freedom (and thus very heavy tails) and exhibit very high values of tail dependence coefficients. Hence the considered bond indices do not represent suitable components of a well-diversified investment portfolio. On the other hand, they could make good candidates for underlying assets of derivative instruments.

Predictive and Descriptive Qualities of Different Classes of Models for Parallel Economic Development of Selected EU-Countries

In this paper, we extend and modify our modelling [presented in the conference paper (Komornik and Komornikova, Predictive and descriptive models of mutual development of economic growth of Germany and selected non-traditional EU countries. In: ITISE 2015, International Work-Conference on Time Series, pp. 55–64. Copicentro Granada S.L, 2015)] of the parallel development of GDP of Germany (as the strongest EU economy), the so-called V4 countries (Poland, the Czech Republic, Hungary, Slovakia) and Greece (as the most problematic EU economy). Unlike in Komornik and Komornikova (Predictive and descriptive models of mutual development of economic growth of Germany and selected non-traditional EU countries. In: ITISE 2015, International Work-Conference on Time Series, pp. 55–64. Copicentro Granada S.L, 2015), we analyse the data provided by OECD (freely available from http://stats.oecd.org/index.aspx?queryid=218) that are expressed in USD (using the expenditure approach) and covering a longer time interval than our former data from EUROSTAT (http://appsso.eurostat.ec.europa.eu/nui/show.do?wai=true&data-set=namq_10_gdp) (expressed in EUR using the output approach). The best predictive quality models were found in the class of multivariate TAR (Threshold Autoregressive) models with aggregation functions’ type thresholds. On the other hand, the best descriptive quality models were found in the competing classes of one-dimensional MSW (Markov Switching) and STAR (Smooth Transition Autoregressive) models.

Modelling multiple REIT indices using TAR models based on aggregation functions

The aim of this paper is to compare descriptive and predictive qualities of multivariate TAR models with threshold variables obtained via aggregation functions versus one-dimensional TAR models with endogenous as well as exogenous threshold variables. Time series of REIT indexes of 5 selected G7 countries (USA, Japan, Great Britain, France, Canada) were modelled. They manifest similar behaviour in the considered time period, January 1, 2000-May 8, 2012, divided into 3 sub-periods determined by the recent global financial markets crisis (July 1, 2008-April 30, 2009). The multivariate TAR models with threshold variables constructed via aggregation functions have in all cases better descriptive properties and in most cases they also show better prediction properties. A new subclass of those models, based on the OMA type of aggregation functions, exhibit promising properties both with respect to their descriptive and predictive performance.

Application of Regime-Switching Models of Time Series with Cubic Spline Transition Function

A new class of Smooth Transition Autoregressive(STAR) models, based on cubic spline type transition functions, has been recently introduced by the authors of this paper and subjected to comparison with models based on the traditional logistic functions. A very high degree of similarity between these two classes of models has been demonstrated.