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Approximate polynomial expansion for joint density

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Pommeret D.

Applicationes Mathematicae

2005

tom 32

nr 1

57-67

Let (X,Y) be a random vector with joint probability measure σ and with margins μ and ν. Let $(Pₙ)_{n∈ℕ}$ and $(Qₙ)_{n∈ℕ}$ be two bases of complete orthonormal polynomials with respect to μ and ν, respectively. Under integrability conditions we have the following polynomial expansion: $σ(dx,dy) = ∑_{n,k∈ℕ} ϱ_{n,k} Pₙ(x)Q_k(y) μ(dx)ν(dy)$. In this paper we consider the problem of changing the margin μ into μ̃ in this expansion. That is the case when μ is the true (or estimated) margin and μ̃ is its approximation. It is shown that a new joint probability with new margins is obtained. The first margin is μ̃ 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.

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1 Applicationes Mathematicae

1 Pommeret D.

1 2005

Wyniki wyszukiwania

Approximate polynomial expansion for joint density