Statistical asymptotic expansions

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Edgeworth series

Let a distribution to be approximated be the distribution of standardized sums(1)In the Charlier series, take the component random variables identically distributed with mean , variance , and higher cumulants for . Also, take the developing function as the standard normal distribution function , so we have(2)(3)(4)Then the Edgeworth series is obtained by collecting terms to obtain the asymptotic expansion of the characteristic function of the form(5)where is a polynomial of degree with coefficients depending on the cumulants of orders 3 to . If the powers of are interpreted as derivatives, then the distribution function expansion is given by(6)(Wallace 1958). The first few terms of this expansion are then given by(7)Cramér (1928) proved that this series is uniformly valid in ...

Charlier series

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(2)(Wallace 1958). Integrating by parts gives as the characteristic function of , so the formal identity corresponds pairwise to the identity(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..

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