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I am looking for references pertaining to, and/or help in deriving, generalizations of the Gram-Charlier and Edgeworth series for non-Gaussian reference probability distributions.


I would like to outline here a sketch of an approach I am taking in deriving a generalization, which could help explain the problems I am having. I am hoping that this might remind someone of something useful which I should be looking into for this problem.

I am considering a generalization of the exposition in, e.g. Kendall's Advanced Theory of Statistics, vol. 1, Section 6.20 and thereabouts. Consider two formal series expansions of a probability density function $f(x)$ in orthogonal polynomials of the forms

$$ f(x) = \sum_{n=0}^\infty c_n \phi_n(x) w(x) $$ $$ f(x) = g\left(\sum_{n=0}^\infty k_n D_n\right) w(x) $$

where $\phi_n$ are the orthogonal polynomials related to the weight function $w$ and by construction we have $c_n$ is the expectation of the $n$th orthogonal polynomial, i.e. $$ c_n = \mathbb E_f [\phi_n] = \int_{\mathbb R} \phi_n (x) f(x) dx$$

and in the second equation $D_n$ is some differential operator.

The usual Gram-Charlier or Edgeworth series is recovered when $w$ is the standard Gaussian, $g(z) = e^z$, $D_n = (-d/dx)^n $ and $\phi_n$ are the Hermite polynomials with suitable normalization. Then the coefficients $k_n$ are just

$$k_n = \frac{\kappa_n}{n!} $$

where $\kappa_n$ are the cumulants of the probability distribution $f$.

My first attempt at a generalization is to ask for what suitable $g(z)$ and $D$ to use given $w$, and what these coefficients $k_n$ become. There are standard techniques for construction the orthogonal polynomial family $\phi_n$ for a given $w$ so this determines the coefficients $c_n$. But keeping $g$ and $D$ unchanged from the Gaussian case seems to cause problems for distributions with compact support.

Consider the semicircle as a concrete example. The corresponding orthogonal polynomials are Chebyshev polynomials of the second kind $U_n$. However if we expand the second equation to just the first two terms,

$$ f(x) = \left(k_0 - k_1 \frac d {dx}\right) w(x)$$

the leading term involves $w^\prime$ which blows up at the edges of the semicircle. This does not seem to produce a useful series in the sense of being able to describe small corrections to $w$. (I realize that G-C and Edgeworth themselves don't always converge nicely either; I am ignoring this for now.) However, the Rodrigues formula for the Chebyshev polynomials appear to suggest that $D_n$ = (some constant depending on $n$) $\times \frac{d^n}{dx^n} (1-x^2)^n$ is more useful, as for the semicircle $w(x) \propto \sqrt{1-x^2}$,

$$D_n w(x) = U_n(x) w(x) $$

However, I got stuck trying to figure out $g(z)$ and $k_n$ should be.

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