This paper develops both univariate and multivariate distributions based on Gram-Charlier and Edgeworth expansions, attempting to ensure non negativity by exploiting the orthogonal properties of the Hermite polynomials. The article motivates the problems underlying some specifications (in particular those involving other conditional moments beyond the variance) and provides empirical examples c...

We show that multiple orthogonal polynomials for r measures (μ1, . . . , μr) satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices ~n ± ~ej, where ~ej are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal ...

In this paper we study an extension of the Gram–Charlier (GC) density in Jondeau and Rockinger (2001) which consists a Gallant Nychka (1987) transformation to ensure positivity without parameter restrictions. We derive its parametric properties such as unimodality, cumulative distribution, higher-order moments, truncated closed-form expressions for expected shortfall (ES) lower partial moments....

In this paper we introduce a notion of duality for matrix valued orthogonal polynomials with respect to measure supported on the nonnegative integers. We show that dual families are closely related certain difference operators acting polynomials. These belong so called Fourier algebras, which play key role in construction families. order illustrate duality, describe family Charlier type explici...