Approximate polynomial expansion for joint density
Volume 32 / 2005
Abstract
Let $(X,Y)$ be a random vector with joint probability measure $\sigma $ and with margins $\mu $ and $\nu $. Let $(P_n)_{n\in {\Bbb N}}$ and $(Q_n)_{n\in {\Bbb N}}$ be two bases of complete orthonormal polynomials with respect to $\mu $ and $\nu $, respectively. Under integrability conditions we have the following polynomial expansion: $$ \sigma (dx,dy) = \displaystyle \sum _{n,k\in {\Bbb N}} \varrho _{n,k} P_n(x)Q_k(y) \mu (dx)\nu (dy). $$ In this paper we consider the problem of changing the margin $\mu $ into $\tilde {\mu }$ in this expansion. That is the case when $\mu $ is the true (or estimated) margin and $\tilde {\mu }$ is its approximation. It is shown that a new joint probability with new margins is obtained. The first margin is $\tilde {\mu }$ and the second one is expressed using connections between orthonormal polynomials. These transformations are compared with those obtained by the Sklar Theorem via copulas. A bound for the distance between the new joint distribution and its parent is proposed. Different cases are illustrated.
