Gram-Charlier series
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
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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 [1] and C.V.L. Charlier [2] in their study of functions of the form
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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
[1] | 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 |
[2] | C.V.L. Charlier, "Frequency curves of type ![]() |
[3] | A.K. Mitropol'skii, "Curves of distributions" , Leningrad (1960) (In Russian) |
Comments
Cf. also Edgeworth series.
References
[a1] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) pp. Sect. 17.6 |
Gram-Charlier series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gram-Charlier_series&oldid=18126