Construction of bivariate distributions and statistical dependence operations

Resumen

Dependence between random variables is studied at various levels in the first part, while the last two chapters are devoted to the construction of bivariate distributions via principal components. Chapter 1 of Preliminaries is devoted to general dependence concepts (Fréchet classes, copulas, and parametric families of distributions). In Chapter 2, we generalize the union and intersection operations of two distance matrices to symmetric nonnegative definite matrices. These operations are shown to be useful in the geometric interpretation of Related Metric Scaling (RMS ), and possibly in other approaches of Multivariate Analysis. They show relevant properties that are studied in this chapter. The behaviour of the operations is, in some way, analogous to that presented by the intersection and union between vector spaces; in particular, we prove that the intersection of orthogonal matrices is the null matrix, while the union is the direct sum of the matrices. Matrices that share their eigenvectors form an equivalence class, and a partial order relation is defined. This class is closed for the union and intersection operations. A continuous extension of these operations is presented in Chapter 3. Infinite matrices are studied in the context of bounded integral operators and numerical kernels. We put the basis for extending RMS to continuous random variables and, hence, infinite matrices. The starting point is Mercer's Theorem, which ensures the existence of an orthogonal expansion of the covariance kernel K (s, t) = min {F (s) , F (t)} - F (s) F (t), where F is the cumulative distribution function of each marginal variable. The sets of eigenvalues and eigenfunctions of K, whose existence is ensured by the cited theorem, allow us to define a product between symmetric and positive (semi)definite kernels, and, further, to define the intersection and the union between them. Results obtained in the discrete instance are extended in this chapter to continuous variables, with examples. Such covariance kernels (symmetric and positive definite) are associated with symmetric and positive quadrant dependent (PQD) bivariate distributions. Covariance between functions of bounded variation defined on the range of some random variables, joined by distributions of this type, can be computed by means of their cumulative distribution functions. In Chapter 4, further consequences are obtained, especially some relevant relations between the covariance and the Fréchet bounds, with a number of results that can be useful in the characterization of independence as well as in testing goodness-of-fit. The intersection of two kernels (defined in Chapter 3) is a particular instance of the covariance between functions. Covariance is a quasiinner product defined through the joint distribution of the variables involved. A measure of affinity between functions with respect to H is defined, and also studied. In Chapter 5, from the concept of affinity between functions via an extension of the covariance, we define the dimension of a distribution, we relate it to the diagonal expansion and find the dimension for some parametric families. Diagonal expansions of bivariate distributions (Lancaster) allows us to construct bivariate distributions. It has proved to be adequate for constructing Markov processes, and has also been applied to engineering problems among other uses. This method has been generalized using the principal dimensions of each marginal variable that are, by construction, canonical variables. We introduce in Chapter 6 the theoretical foundations of this method. In Chapter 7 we study the bivariate, symmetric families obtained when the marginals are Uniform on (0, 1), Exponential with mean 1, standard Logistic, and Pareto (3,1). Conditions for the bivariate density, first canonical correlation and maximum correlation of each family of densities are given in some cases. The corresponding copulas are obtained. " RESUM: Al Capítol 1 de Preliminars es revisen conceptes de dependència generals (classes de Fréchet, còpules, i famílies paramètriques de distribucions). Al Capítol 2, generalitzem les operacions unió i intersecció de dues matrius de distàncies a matrius simètriques semidefinides positives qualssevol. Aquestes operacions shan mostrat dutilitat en la interpretació geomètrica del Related Metric Scaling (RMS), i possiblement en altres tècniques dAnàlisi Multivariant. Sestudien llur propietats que són similars, en alguns aspectes, a les de la unió i intersecció de subespais vectorials. Al Capítol 3 es presenta una extensió al continuu daquestes operacions, mitjançant matrius infinites en el context dels operadors integrals acotats i nuclis numèrics. Sestableix la base per a extendre el RMS a variables contínues i, per tant, a matrius infinites. Es parteix del Teorema de Mercer el qual assegura lexistència duna expansió ortogonal del nucli de la covariança K (s, t) = min {F (s), F (t)} - F (s) F (t), on F és la funció de distribució de cada variable marginal. Els conjunts de valors i funcions pròpies daquest nucli ens permeten definir un producte entre nuclis i la intersecció i unió entre nuclis simètrics semidefinits positius. Tals nuclis de covariança sassocien amb distribucions bivariants també simètriques i amb dependència quadrant positiva (PQD). El producte de dos nuclis és un cas particular de covariança entre funcions, que es pot obtenir a partir de les distribucions conjunta i marginals, com sestudia al Capítol 4 per a funcions de variació afitada, fixada la distribució bivariant H. Sobtenen interessants relacions amb les cotes de Fréchet. Aquesta covariança entre funcions és un producte quasiescalar a lespai de funcions de variació afitada i permet definir una mesura dafinitat. Al Capítol 5 aquesta H-afinitat sutilitza per definir la dimensió duna distribució. Les components principals duna variable (Capítol 6) sutilitzen com a variables canòniques a lexpansió diagonal de Lancaster (Capítol 7 i últim) per a construïr distribucions bivariants amb marginals Uniformes al (0,1), Exponencial de mitjana 1, Logística estàndard, i Pareto (3,1). Sobtenen condicions per la densitat bivariant, correlacions canòniques i correlació màxima per cada família. Sobtenen les còpules corresponents."