In this paper we introduce some new copulas emerging from shock models. It was shown in [21] that... more In this paper we introduce some new copulas emerging from shock models. It was shown in [21] that reflected maxmin copulas (RMM for short) are not just some specific singular copulas; they contain many important absolutely continuous copulas including the negative quadrant dependent "half" of the Eyraud-Farlie-Gumbel-Morgenstern class. The main goal of this paper is to develop the RMM copulas with dependent endogenous shocks and give evidence that RMM copulas may exhibit some characteristics better than the original maxmin copulas (MM for short): (1) An important evidence for that is the iteration procedure of the RMM transformation which we prove to be always convergent and we give many properties of it that are useful in applications. (2) Using this result we find also the limit of the iteration procedure of the MM transformation thus answering a question proposed in [10]. (3) We give the multivariate dependent RMM copula that compares to the MM version given in [10]. In all our copulas the idiosyncratic and systemic shocks are combined via asymmetric linking functions as opposed to Marshall copulas where symmetric linking functions are used.
The omnipotence of copulas when modeling dependence given marginal distributions in a multivariat... more The omnipotence of copulas when modeling dependence given marginal 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 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 p-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 Omladič and Stopar (2019) thus raising the importance of our results. The main technical difficulty
Bivariate imprecise copulas have recently attracted substantial attention. However, the multivari... more Bivariate imprecise copulas have recently attracted substantial attention. However, the multivariate case seems still to be a "blank slate". It is then natural that this idea be tested first on shock model induced copulas, a family which might be the most useful in various applications. We investigate a model in which some of the shocks are assumed imprecise and develop the corresponding set of copulas. In the Marshall's case we get a coherent set of distributions and a coherent set of copulas, where the bounds are naturally corresponding to each other. The situation with the other two groups of multivariate imprecise shock model induced copulas, i.e., the maxmin and the the reflected maxmin (RMM) copulas, is substantially more involved, but we are still able to produce their properties. These are the main results of the paper that serves as the first step into a theory that should develop in this direction. In addition, we unfold the theory of bivariate imprecise RMM copulas that has not yet been done before.
In this paper we present a surprisingly general extension of the main result of a paper that appe... more 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 quasidistributions based on an idea presented in a paper by R. Nelsen with collaborators; (2) starting from what is called (bivariate) p-box in the above mentioned paper we propose some new techniques based on what we call restricted (bivariate) p-box; and (3) a substantial extension of a theory on coherent imprecise copulas developed by M. Omladič and N. Stopar in a previous paper in order to handle coherence of restricted (bivariate) p-boxes. A side result of ours of possibly even greater importance is the following: Every bivariate distribution whether obtained on a usual σ-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.
We discuss avoidance of sure loss and coherence results for semicopulas and standardized function... more We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value 1 at p1, 1,. .. , 1q. We characterize the existence of k-increasing n-variate function C fulfilling A ď C ď B for standardized n-variate functions A, B and discuss the method for constructing this function. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when A respectively B coincides with the pointwise infimum respectively supremum of the set of all k-increasing n-variate functions C fulfilling A ď C ď B.
An investigation is presented of how a comprehensive choice of five most important measures of co... more An investigation is presented of how a comprehensive choice of five most important measures of concordance (namely Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta, and their weaker counterpart Spearman's footrule) relate to non-exchangeability, i.e., asymmetry on copulas. Besides these results, the method proposed also seems to be new and may serve as a raw model for exploration of the relationship between a specific property of a copula and some of its measures of dependence structure, or perhaps the relationship between various measures of dependence structure themselves.
We determine all homomorphisms of the multiplicative group C * into GL n (C). As a consequence we... more We determine all homomorphisms of the multiplicative group C * into GL n (C). As a consequence we get a complete description of homomorphisms of GL n (C) into GL m (C), m < n. We also obtain the general form of multiplicative maps g : C → M n (C).
We consider several questions on spaces of nilpotent matrices. We present sufficient conditions f... more We consider several questions on spaces of nilpotent matrices. We present sufficient conditions for triangularizability and give examples of irreducible spaces. We give a necessary and sufficient condition, in terms of the trace, for all linear combinations of a given set of operators to be nilpotent. We also consider the question of the dimension of a space _z? of nilpotents on 5". In particular, we give a simple new proof of a theorem due to M. Gerstenhaber concerning the maximal dimension of such spaces. 1.
Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { f (x) + ... more Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { f (x) + f (y) , f (x) − f (y) } = { x + y , x − y }, x, y ∈ X, if and only if f is phase equivalent to a surjective linear isometry, that is, f = σU , where U : X → Y is a surjective linear isometry and σ : X → {−1, 1}. This is a Wigner's type result for real normed spaces.
Copula is a useful tool that captures the dependence structure among random variables. In practic... more Copula is a useful tool that captures the dependence structure among random variables. In practice, it is an important question which copula to choose depending on the given data and stochastic assumptions on the model in order to achieve an appropriate interpretation of the data at hand. This paper intends to help a practitioner to make a better decision about that. We concentrate on the study of the lack of exchangeability, a copulas' attribute closely studied only recently. The main non-exchangeability measure µ ∞ for a family of copulas is the supremum of the differences |C(x, y) − C(y, x)| over all (x, y) and all copulas C in the family. We give the sharp bound of µ ∞ for the families of Marshall copulas, maxmin and reflected maxmin copulas (i.e. the main shock-model based copulas) as well as the families of positively and of negatively quadrant dependent copulas. A major contribution of this paper is also exact calculation of the maximal asymmetry function on each of the particular families of copulas. When restricted to special families of copulas considered, it helps us finding the sharp bound of µ ∞ for each of the given families. And even more importantly, it helps us giving a stochastic interpretation of the extremal copulas and examples of shock models where the maximal asymmetry is attained.
Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { f (x) + ... more Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { f (x) + f (y) , f (x) − f (y) } = { x + y , x − y }, x, y ∈ X, if and only if f is phase equivalent to a surjective linear isometry, that is, f = σU , where U : X → Y is a surjective linear isometry and σ : X → {−1, 1}. This is a Wigner's type result for real normed spaces.
When choosing the right copula for our data a key point is to distinguish the family that describ... more When choosing the right copula for our data a key point is to distinguish the family that describes it at the best. In this respect, a better choice of the copulas could be obtained through the information about the (non)symmetry of the data. Exchangeability as a probability concept (first next to independence) has been studied since 1930's, copulas have been studied since 1950's, and even the most important class of copulas from the point of view of applications, i.e. the ones arising from shock models s.a. Marshall's copulas, have been studied since 1960's. However, the point of non-exchangeability of copulas was brought up only in 2006 and has been intensively studied ever since. One of the main contributions of this paper is the maximal asymmetry function for a family of copulas. We compute this function for the major families of shock-based copulas, i.e. Marshall, maxmin and reflected maxmin (RMM for short) copulas and also for some other important families. We ...
Let ϕ be a linear functional of rank one acting on an irreducible semigroup S of operators on a f... more Let ϕ be a linear functional of rank one acting on an irreducible semigroup S of operators on a finite-or infinite-dimensional Hilbert space. It is a well-known and simple fact that the range of ϕ cannot be a singleton. We start a study of possible finite ranges for such functionals. In particular, we prove that in certain cases, the existence of a single such functional ϕ with a two-element range yields valuable information on the structure of S.
The maximal dimension of commutative subspaces of M n (C) is known. So is the structure of such a... more The maximal dimension of commutative subspaces of M n (C) 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 V is a subspace of M n (C) and k is an integer less than n, such that for every pair A and B of members of V , the rank of the commutator AB − BA is at most k, then how large can the dimension of V be? If this maximum is achieved, can we determine the structure of V ? We answer the first question. We also propose a conjecture on the second question which implies, in particular, that such a subspace V has to be an algebra, just as in the known case of k = 0. We prove the proposed structure of V if it is already assumed to be an algebra.
Copula models have become popular in different applications, including modeling shocks, in view o... more Copula models have become popular in different applications, including modeling shocks, in view of their ability to describe better the dependence concepts in stochastic systems. The class of maxmin copulas was recently introduced by Omladi\v{c} and Ru\v{z}i\'{c}. It extends the well known classes of Marshall-Olkin and Marshall copulas by allowing the external shocks to have different effects on the two components of the system. By a reflection (flip) in one of the variables we introduce a new class of bivariate copulas called reflected maxmin (RMM) copulas. We explore their properties and show that symmetric RMM copulas relate to general RMM copulas similarly as do semilinear copulas relate to Marshall copulas. We transfer that relation also to maxmin copulas. We also characterize possible diagonal functions of symmetric RMM copulas.
In the first part of the paper we give a characterization of groups generated by elements of fixe... more In the first part of the paper we give a characterization of groups generated by elements of fixed prime order p. In the second part we study the group G (p) n of n×n matrices with the p-th power of the determinant equal to 1 over a field F containing a primitive p-th root of 1. It is known that the group G (2) n of n × n matrices of determinant +−1 over a field F and the group SLn(F ) are generated by their involutions and that each element in these groups is a product of four involutions. We consider some subgroups G of G (p) n and study the following problems: Is G generated by its elements of order p ? If so, is every element of G a product of k elements of order p for some fixed integer k ? We show that G (p) n and SLn(F ) are generated by their elements of order p and that the bound k exists and is equal to 4. We show that every universal p-Coxeter group has faithful two-dimensional representations over many fields F (including R and C). For a universal p-Coxeter group of rank...
Proceedings of the American Mathematical Society, 2000
It is proved that every invertible bounded linear operator on a complex infinite-dimensional Hilb... more It is proved that every invertible bounded linear operator on a complex infinite-dimensional Hilbert space is a product of five n n -th roots of the identity for every n > 2 n > 2 . For invertible normal operators four factors suffice in general.
In this paper we introduce some new copulas emerging from shock models. It was shown in [21] that... more In this paper we introduce some new copulas emerging from shock models. It was shown in [21] that reflected maxmin copulas (RMM for short) are not just some specific singular copulas; they contain many important absolutely continuous copulas including the negative quadrant dependent "half" of the Eyraud-Farlie-Gumbel-Morgenstern class. The main goal of this paper is to develop the RMM copulas with dependent endogenous shocks and give evidence that RMM copulas may exhibit some characteristics better than the original maxmin copulas (MM for short): (1) An important evidence for that is the iteration procedure of the RMM transformation which we prove to be always convergent and we give many properties of it that are useful in applications. (2) Using this result we find also the limit of the iteration procedure of the MM transformation thus answering a question proposed in [10]. (3) We give the multivariate dependent RMM copula that compares to the MM version given in [10]. In all our copulas the idiosyncratic and systemic shocks are combined via asymmetric linking functions as opposed to Marshall copulas where symmetric linking functions are used.
The omnipotence of copulas when modeling dependence given marginal distributions in a multivariat... more The omnipotence of copulas when modeling dependence given marginal 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 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 p-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 Omladič and Stopar (2019) thus raising the importance of our results. The main technical difficulty
Bivariate imprecise copulas have recently attracted substantial attention. However, the multivari... more Bivariate imprecise copulas have recently attracted substantial attention. However, the multivariate case seems still to be a "blank slate". It is then natural that this idea be tested first on shock model induced copulas, a family which might be the most useful in various applications. We investigate a model in which some of the shocks are assumed imprecise and develop the corresponding set of copulas. In the Marshall's case we get a coherent set of distributions and a coherent set of copulas, where the bounds are naturally corresponding to each other. The situation with the other two groups of multivariate imprecise shock model induced copulas, i.e., the maxmin and the the reflected maxmin (RMM) copulas, is substantially more involved, but we are still able to produce their properties. These are the main results of the paper that serves as the first step into a theory that should develop in this direction. In addition, we unfold the theory of bivariate imprecise RMM copulas that has not yet been done before.
In this paper we present a surprisingly general extension of the main result of a paper that appe... more 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 quasidistributions based on an idea presented in a paper by R. Nelsen with collaborators; (2) starting from what is called (bivariate) p-box in the above mentioned paper we propose some new techniques based on what we call restricted (bivariate) p-box; and (3) a substantial extension of a theory on coherent imprecise copulas developed by M. Omladič and N. Stopar in a previous paper in order to handle coherence of restricted (bivariate) p-boxes. A side result of ours of possibly even greater importance is the following: Every bivariate distribution whether obtained on a usual σ-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.
We discuss avoidance of sure loss and coherence results for semicopulas and standardized function... more We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, 1-increasing functions with value 1 at p1, 1,. .. , 1q. We characterize the existence of k-increasing n-variate function C fulfilling A ď C ď B for standardized n-variate functions A, B and discuss the method for constructing this function. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when A respectively B coincides with the pointwise infimum respectively supremum of the set of all k-increasing n-variate functions C fulfilling A ď C ď B.
An investigation is presented of how a comprehensive choice of five most important measures of co... more An investigation is presented of how a comprehensive choice of five most important measures of concordance (namely Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta, and their weaker counterpart Spearman's footrule) relate to non-exchangeability, i.e., asymmetry on copulas. Besides these results, the method proposed also seems to be new and may serve as a raw model for exploration of the relationship between a specific property of a copula and some of its measures of dependence structure, or perhaps the relationship between various measures of dependence structure themselves.
We determine all homomorphisms of the multiplicative group C * into GL n (C). As a consequence we... more We determine all homomorphisms of the multiplicative group C * into GL n (C). As a consequence we get a complete description of homomorphisms of GL n (C) into GL m (C), m < n. We also obtain the general form of multiplicative maps g : C → M n (C).
We consider several questions on spaces of nilpotent matrices. We present sufficient conditions f... more We consider several questions on spaces of nilpotent matrices. We present sufficient conditions for triangularizability and give examples of irreducible spaces. We give a necessary and sufficient condition, in terms of the trace, for all linear combinations of a given set of operators to be nilpotent. We also consider the question of the dimension of a space _z? of nilpotents on 5". In particular, we give a simple new proof of a theorem due to M. Gerstenhaber concerning the maximal dimension of such spaces. 1.
Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { f (x) + ... more Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { f (x) + f (y) , f (x) − f (y) } = { x + y , x − y }, x, y ∈ X, if and only if f is phase equivalent to a surjective linear isometry, that is, f = σU , where U : X → Y is a surjective linear isometry and σ : X → {−1, 1}. This is a Wigner's type result for real normed spaces.
Copula is a useful tool that captures the dependence structure among random variables. In practic... more Copula is a useful tool that captures the dependence structure among random variables. In practice, it is an important question which copula to choose depending on the given data and stochastic assumptions on the model in order to achieve an appropriate interpretation of the data at hand. This paper intends to help a practitioner to make a better decision about that. We concentrate on the study of the lack of exchangeability, a copulas' attribute closely studied only recently. The main non-exchangeability measure µ ∞ for a family of copulas is the supremum of the differences |C(x, y) − C(y, x)| over all (x, y) and all copulas C in the family. We give the sharp bound of µ ∞ for the families of Marshall copulas, maxmin and reflected maxmin copulas (i.e. the main shock-model based copulas) as well as the families of positively and of negatively quadrant dependent copulas. A major contribution of this paper is also exact calculation of the maximal asymmetry function on each of the particular families of copulas. When restricted to special families of copulas considered, it helps us finding the sharp bound of µ ∞ for each of the given families. And even more importantly, it helps us giving a stochastic interpretation of the extremal copulas and examples of shock models where the maximal asymmetry is attained.
Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { f (x) + ... more Let X and Y be real normed spaces and f : X → Y a surjective mapping. Then f satisfies { f (x) + f (y) , f (x) − f (y) } = { x + y , x − y }, x, y ∈ X, if and only if f is phase equivalent to a surjective linear isometry, that is, f = σU , where U : X → Y is a surjective linear isometry and σ : X → {−1, 1}. This is a Wigner's type result for real normed spaces.
When choosing the right copula for our data a key point is to distinguish the family that describ... more When choosing the right copula for our data a key point is to distinguish the family that describes it at the best. In this respect, a better choice of the copulas could be obtained through the information about the (non)symmetry of the data. Exchangeability as a probability concept (first next to independence) has been studied since 1930's, copulas have been studied since 1950's, and even the most important class of copulas from the point of view of applications, i.e. the ones arising from shock models s.a. Marshall's copulas, have been studied since 1960's. However, the point of non-exchangeability of copulas was brought up only in 2006 and has been intensively studied ever since. One of the main contributions of this paper is the maximal asymmetry function for a family of copulas. We compute this function for the major families of shock-based copulas, i.e. Marshall, maxmin and reflected maxmin (RMM for short) copulas and also for some other important families. We ...
Let ϕ be a linear functional of rank one acting on an irreducible semigroup S of operators on a f... more Let ϕ be a linear functional of rank one acting on an irreducible semigroup S of operators on a finite-or infinite-dimensional Hilbert space. It is a well-known and simple fact that the range of ϕ cannot be a singleton. We start a study of possible finite ranges for such functionals. In particular, we prove that in certain cases, the existence of a single such functional ϕ with a two-element range yields valuable information on the structure of S.
The maximal dimension of commutative subspaces of M n (C) is known. So is the structure of such a... more The maximal dimension of commutative subspaces of M n (C) 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 V is a subspace of M n (C) and k is an integer less than n, such that for every pair A and B of members of V , the rank of the commutator AB − BA is at most k, then how large can the dimension of V be? If this maximum is achieved, can we determine the structure of V ? We answer the first question. We also propose a conjecture on the second question which implies, in particular, that such a subspace V has to be an algebra, just as in the known case of k = 0. We prove the proposed structure of V if it is already assumed to be an algebra.
Copula models have become popular in different applications, including modeling shocks, in view o... more Copula models have become popular in different applications, including modeling shocks, in view of their ability to describe better the dependence concepts in stochastic systems. The class of maxmin copulas was recently introduced by Omladi\v{c} and Ru\v{z}i\'{c}. It extends the well known classes of Marshall-Olkin and Marshall copulas by allowing the external shocks to have different effects on the two components of the system. By a reflection (flip) in one of the variables we introduce a new class of bivariate copulas called reflected maxmin (RMM) copulas. We explore their properties and show that symmetric RMM copulas relate to general RMM copulas similarly as do semilinear copulas relate to Marshall copulas. We transfer that relation also to maxmin copulas. We also characterize possible diagonal functions of symmetric RMM copulas.
In the first part of the paper we give a characterization of groups generated by elements of fixe... more In the first part of the paper we give a characterization of groups generated by elements of fixed prime order p. In the second part we study the group G (p) n of n×n matrices with the p-th power of the determinant equal to 1 over a field F containing a primitive p-th root of 1. It is known that the group G (2) n of n × n matrices of determinant +−1 over a field F and the group SLn(F ) are generated by their involutions and that each element in these groups is a product of four involutions. We consider some subgroups G of G (p) n and study the following problems: Is G generated by its elements of order p ? If so, is every element of G a product of k elements of order p for some fixed integer k ? We show that G (p) n and SLn(F ) are generated by their elements of order p and that the bound k exists and is equal to 4. We show that every universal p-Coxeter group has faithful two-dimensional representations over many fields F (including R and C). For a universal p-Coxeter group of rank...
Proceedings of the American Mathematical Society, 2000
It is proved that every invertible bounded linear operator on a complex infinite-dimensional Hilb... more It is proved that every invertible bounded linear operator on a complex infinite-dimensional Hilbert space is a product of five n n -th roots of the identity for every n > 2 n > 2 . For invertible normal operators four factors suffice in general.
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Papers by Matjaž Omladič