Gram-Charlier series

2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

A series defined by the expression

(1)

or

(2)

where is the normalized value of a random variable.

The series (1) is known as the Gram–Charlier series of type ; here

is the -th derivative of , which can be represented as

where are the Chebyshev–Hermite polynomials. The derivatives and the polynomials are orthogonal, owing to which the coefficients can be defined by the basic moments of the given distribution series. If one restricts to the first few terms of the series (1), one obtains

The series (2) is known as a Gram–Charlier series of type ; here

while are polynomials analogous to the polynomials .

If one restricts to the first terms of the series (2), one obtains

Here are the central moments of the distribution, while .

Gram–Charlier series were obtained by J.P. Gram [G] and C.V.L. Charlier [Ch] in their study of functions of the form

These are convenient for the interpolation between the values of the general term of the binomial distribution, where

is the characteristic function of the binomial distribution. The expansion of in powers of yields a Gram–Charlier series of type for , whereas the expansion of in powers of yields a Gram–Charlier series of type .

References

[G]	J.P. Gram, "Ueber die Entwicklung reeller Funktionen in Reihen mittelst der Methode der kleinsten Quadraten" J. Reine Angew. Math. , 94 (1883) pp. 41–73
[Ch]	C.V.L. Charlier, "Frequency curves of type in heterograde statistics" Ark. Mat. Astr. Fysik , 9 : 25 (1914) pp. 1–17
[M]	A.K. Mitropol'skii, "Curves of distributions" , Leningrad (1960) (In Russian)

Comments

Cf. also Edgeworth series.

References

[Cr]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) pp. Sect. 17.6