JEDNOSTKA NAUKOWA KATEGORII A+

# Wydawnictwa / Czasopisma IMPAN / Applicationes Mathematicae / Wszystkie zeszyty

## Approximate polynomial expansion for joint density

### Tom 32 / 2005

Applicationes Mathematicae 32 (2005), 57-67 MSC: Primary 62E10; Secondary 62E17. DOI: 10.4064/am32-1-5

#### Streszczenie

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.

#### Autorzy

• D. PommeretCREST-ENSAI
rue Blaise Pascal
BP 37203
35172 Bruz Cedex, France
e-mail

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

## Przepisz kod z obrazka Odśwież obrazek