Gregory Fredricks - Academia.edu (original) (raw)
Papers by Gregory Fredricks
College Mathematics Journal, 1992
ABSTRACT In this lecture we consider real analytic Cauchy-Riemann structures. We formulate result... more ABSTRACT In this lecture we consider real analytic Cauchy-Riemann structures. We formulate results about their complexifications, their local and global embeddability and their partial ordering. The main reference for this material is Andreotti [1], where the study of Cauchy-Riemann structures is carefully developed from the elementary definitions to the local and global embeddability questions. Let U be an open set in ℝm and consider a system of ℓ complex-valued C∞ vector fields on U,
Springer eBooks, 2002
In this paper we present some of the salient properties of the Bertino family of copulas. We desc... more In this paper we present some of the salient properties of the Bertino family of copulas. We describe the support set of a Bertino copula and show that every Bertino copula is singular. We characterize Bertino copulas in terms of the joint distribution of max(U, V) and min(U, V) when U and V are uniform [0, 1] random variables whose copula is a Bertino copula. Finally, we find necessary and sufficient conditions for a Bertino copula to be extremal.
Journal of Statistical Planning and Inference, Jul 1, 2007
It has long been known that for many joint distributions exhibiting weak dependence, the sample v... more It has long been known that for many joint distributions exhibiting weak dependence, the sample value of Spearman's rho is about 50% larger than the sample value of Kendall's tau. We explain this behavior by showing that for the population analogs of these statistics, the ratio of rho to tau approaches 3/2 as the joint distribution approaches that of two independent random variables. We also find sufficient conditions for determining the direction of the inequality between three times tau and twice rho when the underlying joint distribution is absolutely continuous.
Insurance Mathematics & Economics, Aug 1, 2005
Using an iterated function system, the authors construct families of copulas whose supports are f... more Using an iterated function system, the authors construct families of copulas whose supports are fractals. In particular, they show that the members of one family have supports with arbitrary Hausdorff dimensions in the interval (1,2). They also employ those copulas to construct more general bivariate distribution functions with fractal supports.
Springer eBooks, 1997
ABSTRACT If C is a copula, then its diagonal section is the function δ given by δ(t)=C(t,t). It f... more ABSTRACT If C is a copula, then its diagonal section is the function δ given by δ(t)=C(t,t). It follows that (i) δ(1)=1; (ii) 0≤δ(t 2 )-δ(t 1 )≤2(t 2 -t 1 ) for all t 1 , t 2 in [0,1] with t 1 ≤t 2 ; and (iii) δ(t)≤1 for all t in [0,1]. If δ is any function satisfying (i)–(iii), does there exist a copula C whose diagonal section is δ? We answer this question affirmatively by constructing copulas we call diagonal copulas: K(u,v)=min(u,v,(1/2)[δ(u)+δ(v)]). We also present examples, investigate some dependence properties of random variables which have diagonal copulas, and answer an open question about tail dependence in bivariate copulas by H. Joe [J. Multivariate Anal. 46, No. 2, 262-282 (1993; Zbl 0778.62045)].
Acta et commentationes Universitatis Tartuensis de mathematica, Dec 31, 2008
In this article we prove a conjecture of Salvatore Bertino which centers on a study of the functi... more In this article we prove a conjecture of Salvatore Bertino which centers on a study of the function F (C) = 1/2 0 VC([t, t + 1/2] 2) dt, (1) where C is a copula. This function, which arises in the study of the index of dissimilarity between two random variables (see Bertino 1977), is the integral of the C-volumes of the family of squares of sidelength 1/2 sliding from the lower left to the upper right along the ascending diagonal of the unit square. The conjecture made in 1999 is that the minimum value of F among the Bertino copulas is 1/8, which is the value of F at the Fréchet lower bound. The conjecture is established by showing that a certain family of shuffles of M is dense in the set of Bertino copulas and then solving a problem in weighted graphs, which is equivalent to the conjecture in the restricted case.
Communications in Partial Differential Equations, 1983
Transactions of the American Mathematical Society, 1989
We discuss conditions under which a lens space is 5th order flat.
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1981
l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute ut... more l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
Rendiconti del Seminario matematico e fisico di Milano, Dec 1, 1980
SUNTO.-Dopo aver esposto alcuni risultati recenti sulla risolubilitA locale di sistemi di equazio... more SUNTO.-Dopo aver esposto alcuni risultati recenti sulla risolubilitA locale di sistemi di equazioni differenziali, si definisce i] eomplesso di Cauchy-Riemann indotto su una sottovaretA generica di (~n e si danno eriteri per la non trivialitb dei suoi gruppi di coomologia locale.
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1979
College Mathematics Journal, 1992
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1981
l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute ut... more l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
Insurance: Mathematics and Economics, 2005
Using an iterated function system, the authors construct families of copulas whose supports are f... more Using an iterated function system, the authors construct families of copulas whose supports are fractals. In particular, they show that the members of one family have supports with arbitrary Hausdorff dimensions in the interval (1,2). They also employ those copulas to construct more general bivariate distribution functions with fractal supports.
In this article we prove a conjecture of Salvatore Bertino which centers on a study of the function
Abstract. It has long been known that for many joint distributions, the sample value of Spea rman... more Abstract. It has long been known that for many joint distributions, the sample value of Spea rman’s rho is about 50% larger than the sample value of Kendall’s t u. We explain this behavior by showing that for the population analogs of these statistics, the ratio of rho t tau approaches 3/2 as the distribution approaches that of independent random variables. We also find sufficient conditions for determining the sense of the inequality between three times tau and twice rho when the underlying joint distribution is absolutely co ntinuous. MSC: primary 62H20; secondary 62H05
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1979
L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scien... more L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
L’acces aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scien... more L’acces aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.
College Mathematics Journal, 1992
ABSTRACT In this lecture we consider real analytic Cauchy-Riemann structures. We formulate result... more ABSTRACT In this lecture we consider real analytic Cauchy-Riemann structures. We formulate results about their complexifications, their local and global embeddability and their partial ordering. The main reference for this material is Andreotti [1], where the study of Cauchy-Riemann structures is carefully developed from the elementary definitions to the local and global embeddability questions. Let U be an open set in ℝm and consider a system of ℓ complex-valued C∞ vector fields on U,
Springer eBooks, 2002
In this paper we present some of the salient properties of the Bertino family of copulas. We desc... more In this paper we present some of the salient properties of the Bertino family of copulas. We describe the support set of a Bertino copula and show that every Bertino copula is singular. We characterize Bertino copulas in terms of the joint distribution of max(U, V) and min(U, V) when U and V are uniform [0, 1] random variables whose copula is a Bertino copula. Finally, we find necessary and sufficient conditions for a Bertino copula to be extremal.
Journal of Statistical Planning and Inference, Jul 1, 2007
It has long been known that for many joint distributions exhibiting weak dependence, the sample v... more It has long been known that for many joint distributions exhibiting weak dependence, the sample value of Spearman's rho is about 50% larger than the sample value of Kendall's tau. We explain this behavior by showing that for the population analogs of these statistics, the ratio of rho to tau approaches 3/2 as the joint distribution approaches that of two independent random variables. We also find sufficient conditions for determining the direction of the inequality between three times tau and twice rho when the underlying joint distribution is absolutely continuous.
Insurance Mathematics & Economics, Aug 1, 2005
Using an iterated function system, the authors construct families of copulas whose supports are f... more Using an iterated function system, the authors construct families of copulas whose supports are fractals. In particular, they show that the members of one family have supports with arbitrary Hausdorff dimensions in the interval (1,2). They also employ those copulas to construct more general bivariate distribution functions with fractal supports.
Springer eBooks, 1997
ABSTRACT If C is a copula, then its diagonal section is the function δ given by δ(t)=C(t,t). It f... more ABSTRACT If C is a copula, then its diagonal section is the function δ given by δ(t)=C(t,t). It follows that (i) δ(1)=1; (ii) 0≤δ(t 2 )-δ(t 1 )≤2(t 2 -t 1 ) for all t 1 , t 2 in [0,1] with t 1 ≤t 2 ; and (iii) δ(t)≤1 for all t in [0,1]. If δ is any function satisfying (i)–(iii), does there exist a copula C whose diagonal section is δ? We answer this question affirmatively by constructing copulas we call diagonal copulas: K(u,v)=min(u,v,(1/2)[δ(u)+δ(v)]). We also present examples, investigate some dependence properties of random variables which have diagonal copulas, and answer an open question about tail dependence in bivariate copulas by H. Joe [J. Multivariate Anal. 46, No. 2, 262-282 (1993; Zbl 0778.62045)].
Acta et commentationes Universitatis Tartuensis de mathematica, Dec 31, 2008
In this article we prove a conjecture of Salvatore Bertino which centers on a study of the functi... more In this article we prove a conjecture of Salvatore Bertino which centers on a study of the function F (C) = 1/2 0 VC([t, t + 1/2] 2) dt, (1) where C is a copula. This function, which arises in the study of the index of dissimilarity between two random variables (see Bertino 1977), is the integral of the C-volumes of the family of squares of sidelength 1/2 sliding from the lower left to the upper right along the ascending diagonal of the unit square. The conjecture made in 1999 is that the minimum value of F among the Bertino copulas is 1/8, which is the value of F at the Fréchet lower bound. The conjecture is established by showing that a certain family of shuffles of M is dense in the set of Bertino copulas and then solving a problem in weighted graphs, which is equivalent to the conjecture in the restricted case.
Communications in Partial Differential Equations, 1983
Transactions of the American Mathematical Society, 1989
We discuss conditions under which a lens space is 5th order flat.
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1981
l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute ut... more l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
Rendiconti del Seminario matematico e fisico di Milano, Dec 1, 1980
SUNTO.-Dopo aver esposto alcuni risultati recenti sulla risolubilitA locale di sistemi di equazio... more SUNTO.-Dopo aver esposto alcuni risultati recenti sulla risolubilitA locale di sistemi di equazioni differenziali, si definisce i] eomplesso di Cauchy-Riemann indotto su una sottovaretA generica di (~n e si danno eriteri per la non trivialitb dei suoi gruppi di coomologia locale.
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1979
College Mathematics Journal, 1992
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1981
l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute ut... more l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
Insurance: Mathematics and Economics, 2005
Using an iterated function system, the authors construct families of copulas whose supports are f... more Using an iterated function system, the authors construct families of copulas whose supports are fractals. In particular, they show that the members of one family have supports with arbitrary Hausdorff dimensions in the interval (1,2). They also employ those copulas to construct more general bivariate distribution functions with fractal supports.
In this article we prove a conjecture of Salvatore Bertino which centers on a study of the function
Abstract. It has long been known that for many joint distributions, the sample value of Spea rman... more Abstract. It has long been known that for many joint distributions, the sample value of Spea rman’s rho is about 50% larger than the sample value of Kendall’s t u. We explain this behavior by showing that for the population analogs of these statistics, the ratio of rho t tau approaches 3/2 as the distribution approaches that of independent random variables. We also find sufficient conditions for determining the sense of the inequality between three times tau and twice rho when the underlying joint distribution is absolutely co ntinuous. MSC: primary 62H20; secondary 62H05
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 1979
L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scien... more L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
L’acces aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scien... more L’acces aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.