Edgeworth series

2020 Mathematics Subject Classification: Primary: 60F05 [MSN][ZBL]

The series defined by

| (*)

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Here  is the distribution density of the random variable

(, where  are independent and identically distributed),

is the density of the standard normal distribution, and

The coefficients , , do not depend on  and are polynomials with respect to , where ,  is the variance, and  is the semi-invariant of order  of . In particular, the first terms of the expansion have the form

The coefficients  can also be expressed in terms of the central moments.

The series (*) were introduced by F.Y. Edgeworth [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  in which the remainder has the order of the first discarded term.

#### References[edit]

[E] | F.Y. Edgeworth, "The law of error I" Proc. Cambridge Philos. Soc. , 20 (1905) pp. 36–65

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[C] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) MR0016588 Zbl 0063.01014

#### Comments[edit]

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 [P]. See also [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 [BR]. Extensions to statistics of a more complicated structure, such as -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 [BGZ].

#### References[edit]

[P] | V.V. Petrov, "Sums of independent random variables", Springer (1975) (Translated from Russian) MR0388499 Zbl 0322.60043 Zbl 0322.60042

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[F] | W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971) pp. 135
[BR] | R.N. Bhattacharya, R. Ranga Rao, "Normal approximations and asymptotic expansions", Wiley (1976) MR0436272
[BGZ] | P.J. Bickel, F. Götze, W.R. van Zwet, "The Edgeworth expansion for -statistics of degree two" Ann. Statist., 14 (1986) pp. 1463–1484 MR868312