Josep Fortiana

Distributions With Given Marginals and Statistical Modelling

Preface. List of Participants. On quasi-copulas and metrics C. Alsina. Multivariate survival models incorporating hidden truncation B.C. Arnold, R.J. Beaver. Variation independent parameterizations of multivariate categorical distributions W.P. Bergsma, T. Rudas. A New Proof of Sklars Theorem H. Carley, M.D. Taylor. Diagnonal distributions via orthogonal expansions and tests of independence C.M. Cuadras. Principal Components of the Pareto distribution C.M. Cuadras, Y. Lahlou. Shape of a distribution through the L22-Wasserstein Distance J.A. Cuesta-Albertos, C. Matran Bea, J.M. Rodriguez Rodriguez. Realizable Monotonicity and Inverse Probability Transform J.A. Fill, M. Machida. An Ordering Among Generalized Closeness Criteria R.L. Fountain. The Bertino family of copulas G.A. Fredricks, R.B. Nelsen. Time series models with given interactions R. Fried. Conditions for the asymptotic semiparametric efficiency of an omnibus estimator of dependence parameters in copula models C. Genest, B.J.M. Werker. Maximum correlations and tests of goodness-of-fit A. Grane, J. Fortiana. Which is the right Laplace? S. Kotz. A New Grade Measure of Monotone Multivariate Separability T. Kowalczyk, M. Niewiadomska-Bugai. Some Integration-by-Parts Formulas Involving 2-Copulas X. Li, P. Mikusinski, M.D. Taylor. Bayesian Robustness for Multivariate Problems T.B. Murphy. Concordance and copulas: A survey R.B. Nelsen. Multivariate Archimedean quasi-copulas R.B. Nelsen, J.J. Quesada-Molina, J.A. Rodriguez-Lallena, M. Ubeda-Flores. Some new properties of quasi-copulas R.B. Nelsen, J.J. Quesada-Molina, J.A. Rodriguez-Lallena, M. Ubeda-Flores. Assignment Models for Constrained Marginals and Restricted Markets D.Ramachandran, L. Ruschendorf. Variance minimization and random variables with constant sum L. Ruschendorf, L. Uckelmann. Conditional Expectations and Idempotent Copulae C. Sempi. Existence of Multivariate Distributions with Given Marginals E.-M. Tiit. Topic Index.

Some computational aspects of a distance—based model for prediction

Some new results of a distance—based (DB) model for prediction with mixed variables are presented and discussed. This model can be thought of as a linear model where predictor variables for a response Y are obtained from the observed ones via classic multidimensional scaling. A coefficient is introduced in order to choose the most predictive dimensions, providing a solution to the problem of small variances and a very large number n of observations (the dimensionality increases as n). The problem of missing data is explored and a DB solution is proposed. It is shown that this approach can be regarded as a kind of ridge regression when the usual Euclidean distance is used.

Continuous metric scaling and prediction

Abstract Multidimensional Scaling has been used as a method of ordination rather than prediction. Recently Cuadras and Arenas (Stat. Data Anal. and Inference, 1989; Comm. Stat., 19, 1990; Questiio, 14, 1990) proposed the use of metric scaling in problems of discrimination, regression with mixed variables and non-linear regression. This method uses a distance between observations, hence it is necessary to handle distance matrices of dimension n (the number of individuals). However, for some distances (e.g. the distance based on Gowers similarity coefficient and the square root of the city block distance), the Euclidean dimension increases with n. In this paper, the principal dimensions for the second distance are obtained and studied for a finite set of points, leading to the study of one kind of centrosymmetric matrices. A continuous extension is obtained, i.e. one in which the points are real numbers following a probability distribution. This extension interprets a stochastic process as a continuous Euclidean configuration of points. Then a countable set of principal dimensions is obtained by performing an eigenanalysis on the symmetric kernel of the process and carrying out a suitable transformation of the continuous configuration. Applications to non-linear regression and goodness-of-fit are obtained and some illustrations are given. Keywords and phrases: Principal coordinate analysis; Nonlinear regression; Orthogonal polynomials; Orthogonal decomposition of a stochastic process; Goodness of fit.

Distance-based local linear regression for functional predictors

The problem of nonparametrically predicting a scalar response variable from a functional predictor is considered. A sample of pairs (functional predictor and response) is observed. When predicting the response for a new functional predictor value, a semi-metric is used to compute the distances between the new and the previously observed functional predictors. Then each pair in the original sample is weighted according to a decreasing function of these distances. A Weighted (Linear) Distance-Based Regression is fitted, where the weights are as above and the distances are given by a possibly different semi-metric. This approach can be extended to nonparametric predictions from other kinds of explanatory variables (e.g., data of mixed type) in a natural way.

Visualizing Categorical Data with Related Metric Scaling

Publisher Summary This chapter discusses the visualization of categorical data with related metric scaling. Data that can be likened to distances are common in multivariate statistics where they are often called “dissimilarities.” A dissimilarity matrix is a square, symmetric matrix of nonnegative data and has zeros on its diagonal. The metric scaling, also called the principal coordinate analysis, is a technique that allows construction of a map or Euclidean configuration from a matrix of dissimilarities. Because the same set of distances can be obtained from several Euclidean configurations of points, one of them is selected as the usual metric scaling solution. The main advantage of the metric scaling becomes apparent when a dissimilarity matrix is processed that has not been obtained from actual measurements from a map. The chapter also discusses the basics of the metric scaling, metric scaling graphic representation, and methodology for the metric scaling along with an empirical application.

Selection of Predictors in Distance-Based Regression

Distance-based regression is a prediction method consisting of two steps: from distances between observations we obtain latent variables which, in turn, are the regressors in an ordinary least squares linear model. Distances are computed from actually observed predictors by means of a suitable dissimilarity function. Being generally nonlinearly related with the response, their selection by the usual F tests is unavailable. In this article, we propose a solution to this predictor selection problem by defining generalized test statistics and adapting a nonparametric bootstrap method to estimate their p-values. We include a numerical example with automobile insurance data.

Continuous Extensions of Matrix Formulations in Correspondence Analysis, with Applications to the FGM Family of Distributions

Correspondence analysis (CA) is a method designed to give a graphical representation of a contingency table N and thus to interpret the association between rows and columns. To be specific, correspondence analysis visualises the so-called correspondence matrix P, which is the discrete bivariate density obtained by dividing N by its grand total n: P = (1/n)N. A continuous extension of CA can be obtained by replacing P with a bivariate probability density h(x, y). The marginal densities f(x), g(y) are the continuous counterparts of the row and column margins of P.

A scale-free goodness-of-fit statistic for the exponential distribution based on maximum correlations ☆

Abstract We propose a goodness-of-fit statistic for testing exponentiality based on Hoeffdings maximum correlation. We study its small and large sample properties, we obtain its exact distribution, tables of exact and asymptotic critical values, and some power curves. We compare this statistic with the Gini statistic, the Shapiro–Wilk statistic for exponentiality and the Stephens’ modification of the latter.

A family of matrices, the discretized brownian bridge, and distance-based regression

Abstract The investigation of a distance-based regression model, using a one-dimensional set of equally spaced points as regressor values and √| x − y | as a distance function, leads to the study of a family of matrices which is closely related to a discrete analog of the Brownian-bridge stochastic process. We describe its eigenstructure and several properties, recovering in particular well-known results on tridiagonal Toeplitz matrices and related topics.

Interaction Terms in Distance-Based Regression

We propose a method of including polynomial and interaction terms in Distance-Based Regression (Cuadras and Arenas, 1990), relying on properties of a semi-Hadamard or Khatri-Rao product of matrices. We demonstrate its application to real data examples.