Some orthogonal expansions for the logistic distribution

Contributions to the diagonal expansion of a bivariate copula with continuous extensions

“…This paper extends some results on the eigenanalysis (EA) of a kernel, obtained by Cuadras and Fortiana [15], Cuadras and Lahlou [17], Cuadras et al [16], Cuadras [6,7], and Cuadras and Cuadras [13].…”

“…If the eigenvalue is negative, then we may consider cor(−φ(U), φ(V )). Furthermore, by relating (7), (17) to (9), (13) and using Proposition 1, clearly any canonical function with maximal correlation satisfies cor(φ(U), φ(V )) = λ, where (φ, λ) is an eigenpair of K w.r.t. L. Thus the canonical analysis, for symmetric copulas, reduces to the eigenanalysis of K w.r.t.…”

“…My contributions to this topic began with the EA of min{F (x), F (y)} − F (x)F (y), studied in [15,17], in order to expand X in a series of orthogonal variables, as a continuous extension of multidimensional scaling (MDS) for a particular distance. In a similar way, by using the chi-square distance, a continuous extension of correspondence analysis was developed in [16].…”

On the Covariance between Functions

“…where the convergence is in the mean-square sense (Cuadras and Fortiana, 1995;Cuadras and Lahlou, 2000). Note that h n may not be a monotonic function.…”

“…We assume that the canonical variables are chosen so that the canonical correlations are positive and arranged in descending order. This expansion, usually expressed for densities, is presented below in terms of cdf's; see (6).…”