Approximate polynomial expansion for joint density

D. Pommeret¹

¹ CREST-ENSAI rue Blaise Pascal BP 37203 35172 Bruz Cedex, France

Applicationes Mathematicae, Tome 32 (2005) no. 1, pp. 57-67.

Résumé

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.

Détail

DOI : 10.4064/am32-1-5

Keywords: random vector joint probability measure sigma margins bbb bbb bases complete orthonormal polynomials respect respectively under integrability conditions have following polynomial expansion sigma displaystyle sum bbb varrho q paper consider problem changing margin tilde expansion estimated margin tilde its approximation shown joint probability margins obtained first margin tilde second expressed using connections between orthonormal polynomials these transformations compared those obtained sklar theorem via copulas bound distance between joint distribution its parent proposed different cases illustrated