Carles M. Cuadras

Skewed multivariate models related to hidden truncation and/or selective reporting

The univariate skew-normal distribution was introduced by Azzalini in 1985 as a natural extension of the classical normal density to accommodate asymmetry. He extensively studied the properties of this distribution and in conjunction with coauthors, extended this class to include the multivariate analog of the skew-normal. Arnold et al. (1993) introduced a more general skew-normal distribution as the marginal distribution of a truncated bivariate normal distribution in whichX was retained only ifY satisfied certain constraints. Using this approach more general univariate and multivariate skewed distributions have been developed. A survey of such models is provided together with discussion of related inference questions.

A continuous general multivariate distribution and its properties

Starting from two known continuous univariate distributions, a bivariate distribution is constructed depending on a parameter which measures the degree of stochastic dependence between the two random variables. From the foregoing construction we then pass to a multivariate-type distribution, constructed using only univariate distributions and an association matrix. Some properties of the multivariate and bivariate case are studied.

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.

A distance based regression model for prediction with mixed data

A multiple regression method based on distance analysis and metric scaling is proposed and studied. This method allow us to predict a continuous response variable from several explanatory variables, is compatible with the general linear model and is found to be useful when the predictor variables are both continuous and categorical. Real data examples are given to illustrate the results obtained.

DISTANCE ANALYSIS IN DISCRIMINATION AND CLASSIFICATION USING BOTH CONTINUOUS AND CATEGORICAL VARIABLES

This paper is concerned with the application of distance functions, together with principal coordinate analysis, to some multivariate problems, namely multiple regression, manova and discriminant analysis. The main problem tackled is that of giving a distance-based rule to assign an individual to one of two distinct groups. The discriminant analysis with both continuous and discrete variables, as well as that with binary, ordinal and qualitative data, is also studied. Closed form expressions of distances between individuals are proposed and discussed.

Contributions of empirical and quantile processes to the asymptotic theory of goodness-of-fit tests

This paper analyzes the evolution of the asymptotic theory of goodness-of-fit tests. We emphasize the parallel development of this theory and the theory of empirical and quantile processes. Our study includes the analysis of the main tests of fit based on the empirical distribution function, that is, tests of the Cramér-von Mises or Kolmogorov-Smirnov type. We pay special attention to the problem of testing fit to a location scale family. We provide a new approach, based on the Wasserstein distance, to correlation and regression tests, outlining some of their properties and explaining their limitations.

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.

Correspondence analysis and diagonal expansions in terms of distribution functions

Correspondence analysis on two categorical variables is usually formulated on a two-way contingency table of frequencies. Assuming the variables to be ordinal and re-expressing the canonical correlation analysis, we show that correspondence analysis may also be approached in terms of the cumulative frequencies. A continuous extension is made. It is shown that the diagonal expansion of a bivariate density may be formulated using integral operators on the cumulative distribution function.

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.

Some orthogonal expansions for the logistic distribution

We expand a continuous random variable as a sum of a sequence of un-correlated random variables. These variables are principal components of a Bernoulli process, as well as principal dimensions in continuous metric scaling on a particular distance function. We obtain expansions for the uniform, exponential and logistic distributions. A goodness-of-fit application is given.