A class of formal series expansions in derivatives of a distribution 
which may (but need not) be the normal
distribution function
 |
(1)
|
and moments or other measured parameters. Edgeworth series are known as the Charlier series or Gram-Charlier series. Let 
be the characteristic
function of the function 
, and 
its cumulants. Similarly,
let 
be the distribution to be approximated, 
its characteristic
function, and 
its cumulants. By definition, these quantities are connected
by the formal series
![f(t)=exp[sum_(r=1)^infty(kappa_r-gamma_r)((it)^r)/(r!)]psi(t)](/images/equations/CharlierSeries/NumberedEquation2.svg) |
(2)
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(Wallace 1958). Integrating by parts gives 
as the characteristic
function of 
,
so the formal identity corresponds pairwise to the identity
![F(x)=exp[sum_(r=1)^infty(kappa_r-gamma_r)((-D)^r)/(r!)]Psi(x),](/images/equations/CharlierSeries/NumberedEquation3.svg) |
(3)
|
where 
is the differential operator. The most important
case 
was considered by Chebyshev (1890), Charlier (1905-06), and Edgeworth (1905).
Expanding and collecting terms according to the order of the derivatives gives the so-called Gram-Charlier A-Series, which is identical to the formal expansion of 
in Hermite polynomials. The A-series
converges for functions 
whose tails approach zero faster than 
(Cramér 1925, Wallace 1958, Szegö 1975).
