View source for Edgeworth series

{{MSC|60F05}}

[[Category:Limit theorems]]

The series defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e0350601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e0350602.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e0350603.png" /> is the distribution density of the random variable

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e0350604.png" /></td> </tr></table>

(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e0350605.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e0350606.png" /> are independent and identically distributed),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e0350607.png" /></td> </tr></table>

is the density of the standard [[Normal distribution|normal distribution]], and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e0350608.png" /></td> </tr></table>

The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e0350609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506010.png" />, do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506011.png" /> and are polynomials with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506014.png" /> is the variance, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506015.png" /> is the [[Semi-invariant(2)|semi-invariant]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506017.png" />. In particular, the first terms of the expansion have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506018.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506019.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506020.png" /></td> </tr></table>

The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506021.png" /> can also be expressed in terms of the central moments.

The series (*) were introduced by F.Y. Edgeworth {{Cite|E}}. Their asymptotic properties have been studied by H. Cramér, who has shown that under fairly general conditions the series (*) is the asymptotic expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506022.png" /> in which the remainder has the order of the first discarded term.

====References====
{|
|valign="top"|{{Ref|E}}|| F.Y. Edgeworth, "The law of error I" ''Proc. Cambridge Philos. Soc.'' , '''20''' (1905) pp. 36–65
|-
|valign="top"|{{Ref|C}}|| H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) {{MR|0016588}} {{ZBL|0063.01014}}
|}

====Comments====
The above discussion omits many technical details as well as modern developments.

An excellent account of the theory of Edgeworth expansions for sums of independent random variables is given in {{Cite|P}}. See also {{Cite|F}}, Chapt. XVI for a brief and very smooth introduction to the theory of Edgeworth expansions. The case of sums of independent random vectors is treated in {{Cite|BR}}. Extensions to statistics of a more complicated structure, such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506024.png" />-statistics — which are especially of interest in statistical theory — were studied by many authors over the last 15 years (as of 1988). An important recent contribution in this area is {{Cite|BGZ}}.

====References====
{|
|valign="top"|{{Ref|P}}|| V.V. Petrov, "Sums of independent random variables", Springer (1975) (Translated from Russian) {{MR|0388499}} {{ZBL|0322.60043}} {{ZBL|0322.60042}}
|-
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971) pp. 135
|-
|valign="top"|{{Ref|BR}}|| R.N. Bhattacharya, R. Ranga Rao, "Normal approximations and asymptotic expansions", Wiley (1976) {{MR|0436272}} {{ZBL|}}
|-
|valign="top"|{{Ref|BGZ}}|| P.J. Bickel, F. Götze, W.R. van Zwet, "The Edgeworth expansion for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035060/e03506025.png" />-statistics of degree two" ''Ann. Statist.'', '''14''' (1986) pp. 1463–1484 {{MR|868312}} {{ZBL|}}
|}