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On the optimality of the max-depth and max-rank classifiers for spherical data

Ondřej Vencálek, Houyem Demni, Amor Messaoud, Giovanni C. Porzio (2020)

Applications of Mathematics

The main goal of supervised learning is to construct a function from labeled training data which assigns arbitrary new data points to one of the labels. Classification tasks may be solved by using some measures of data point centrality with respect to the labeled groups considered. Such a measure of centrality is called data depth. In this paper, we investigate conditions under which depth-based classifiers for directional data are optimal. We show that such classifiers are equivalent to the Bayes...

On the order equivalence relation of binary association measures

Mariusz Paradowski (2015)

International Journal of Applied Mathematics and Computer Science

Over a century of research has resulted in a set of more than a hundred binary association measures. Many of them share similar properties. An overview of binary association measures is presented, focused on their order equivalences. Association measures are grouped according to their relations. Transformations between these measures are shown, both formally and visually. A generalization coefficient is proposed, based on joint probability and marginal probabilities. Combining association measures...

On the properties of the Generalized Normal Distribution

Thomas L. Toulias, Christos P. Kitsos (2014)

Discussiones Mathematicae Probability and Statistics

The target of this paper is to provide a critical review and to enlarge the theory related to the Generalized Normal Distributions (GND). This three term (position, scale shape) distribution is based in a strong theoretical background due to Logarithm Sobolev Inequalities. Moreover, the GND is the appropriate one to support the Generalized entropy type Fisher's information measure.

On two matrix derivatives by Kollo and von Rosen.

Heinz Neudecker (2003)

SORT

The article establishes relationships between the matrix derivatives of F with respect to X as introduced by von Rosen (1988), Kollo and von Rosen (2000) and the Magnus-Neudecker (1999) matrix derivative. The usual transformations apply and the Moore-Penrose inverse of the duplication matrix is used. Both X and F have the same dimension.

On uniform tail expansions of bivariate copulas

Piotr Jaworski (2004)

Applicationes Mathematicae

The theory of copulas provides a useful tool for modelling dependence in risk management. The goal of this paper is to describe the tail behaviour of bivariate copulas and its role in modelling extreme events. We say that a bivariate copula has a uniform lower tail expansion if near the origin it can be approximated by a homogeneous function L(u,v) of degree 1; and it is said to have a uniform upper tail expansion if the associated survival copula has a lower tail expansion. In this paper we (1)...

On uniform tail expansions of multivariate copulas and wide convergence of measures

Piotr Jaworski (2006)

Applicationes Mathematicae

The theory of copulas provides a useful tool for modeling dependence in risk management. In insurance and finance, as well as in other applications, dependence of extreme events is particularly important, hence there is a need for a detailed study of the tail behaviour of multivariate copulas. We investigate the class of copulas having regular tails with a uniform expansion. We present several equivalent characterizations of uniform tail expansions. Next, basing on them, we determine the class of...

On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas

Franco Pellerey (2008)

Kybernetika

Let 𝐗 = ( X , Y ) be a pair of exchangeable lifetimes whose dependence structure is described by an Archimedean survival copula, and let 𝐗 t = [ ( X - t , Y - t ) | X > t , Y > t ] denotes the corresponding pair of residual lifetimes after time t , with t ≥ 0 . This note deals with stochastic comparisons between 𝐗 and 𝐗 t : we provide sufficient conditions for their comparison in usual stochastic and lower orthant orders. Some of the results and examples presented here are quite unexpected, since they show that there is not a direct correspondence between univariate...

One singular multivariate linear model with nuisance parameters

Pavla Kunderová (2005)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The multivariate linear model, in which the matrix of the first order parameters is divided into two matrices: to the matrix of the useful parameters and to the matrix of the nuisance parameters, is considered.

Optimal nonlinear transformations of random variables

Aldo Goia, Ernesto Salinelli (2010)

Annales de l'I.H.P. Probabilités et statistiques

In this paper we deepen the study of the nonlinear principal components introduced by Salinelli in 1998, referring to a real random variable. New insights on their probabilistic and statistical meaning are given with some properties. An estimation procedure based on spline functions, adapting to a statistical framework the classical Rayleigh–Ritz method, is introduced. Asymptotic properties of the estimator are proved, providing an upper bound for the rate of convergence under suitable mild conditions....

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