Edgeworth Series
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
1-4-2021
3719
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
Then the Edgeworth series is obtained by collecting terms to obtain the asymptotic expansion of the characteristic function of the form
![f_n(t)=[1+sum_(r=1)^infty(P_r(it))/(n^(r/2))]e^(-t^2/2),](https://mathworld.wolfram.com/images/equations/EdgeworthSeries/NumberedEquation2.gif) |
(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
![f(t)=Psi(t)-(lambda_3Psi^((3))(t))/(6sqrt(n))
+1/n[(lambda_4Psi^((4))(t))/(24)+(lambda_3^2Psi^((6))(t))/(72)]+....](https://mathworld.wolfram.com/images/equations/EdgeworthSeries/NumberedEquation4.gif) |
(7)
|
Cramér (1928) proved that this series is uniformly valid in 
.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 935, 1972.
Charlier, C. V. L. "Über dir Darstellung willkürlicher Funktionen." Ark. Mat. Astr. och Fys. 2, No. 20, 1-35, 1906.
Cramér, H. "On the Composition of Elementary Errors." Skand. Aktuarietidskr. 11, 13-74 and 141-180, 1928.
Edgeworth, F. Y. "The Law of Error." Cambridge Philos. Soc. 20, 36-66 and 113-141, 1905.
Esseen, C. G. "Fourier Analysis of Distribution Functions." Acta Math. 77, 1-125, 1945.
Hsu, P. L. "The Approximate Distribution of the Mean and Variance of a Sample of Independent Variables." Ann. Math. Stat. 16, 1-29, 1945.
Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 107-108, 1951.
Wallace, D. L. "Asymptotic Approximations to Distributions." Ann. Math. Stat. 29, 635-654, 1958.
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