# Gram-Charlier series

(Redirected from Gram–Charlier series)

2010 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

A series defined by the expression

$$\tag{1 } f _ {A} ( x) = \ f ( x) + \sum _ {k = 3 } ^ { n } a _ {k} f ^ { ( k) } ( x)$$

or

$$\tag{2 } f _ {B} ( x) = \ \psi ( x) \sum _ {m = 0 } ^ { n } b _ {m} g _ {m} ( x),$$

where $x$ is the normalized value of a random variable.

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

$$f ( x) = \ { \frac{1}{\sqrt {2 \pi }} } e ^ {- x ^ {2} /2 } ,$$

$f ^ { ( k) }$ is the $k$-th derivative of $f$, which can be represented as

$$f ^ { ( k) } ( x) = \ (- 1) ^ {k} H _ {k} ( x) f ( x),$$

where $H _ {k} ( x)$ are the Chebyshev–Hermite polynomials. The derivatives $f ^ { ( k) }$ and the polynomials $H _ {k}$ are orthogonal, owing to which the coefficients $a _ {k}$ can be defined by the basic moments $r _ {k}$ of the given distribution series. If one restricts to the first few terms of the series (1), one obtains

$$f _ {A} ( x) = \ f ( x) + \frac{r _ {3} }{3!} f ^ { ( 3) } ( x) +$$

$$+ \frac{r _ {4} - 3 }{4! } f ^ { ( 4) } ( x) - \frac{r _ {5} - 10r _ {3} }{5! } f ^ { ( 3) } ( x) + \frac{r _ {4} - 15r _ {4} + 30 }{6! } f ^ { ( 6) } ( x).$$

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

$$\psi ( x) = \ \frac{\lambda ^ {x} }{x!} e ^ {- \lambda } ,\ \ x = 0, 1 \dots$$

while $g _ {m} ( x)$ are polynomials analogous to the polynomials $H _ {k} ( x)$.

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

$$f _ {B} ( x) = \ \frac{\lambda ^ {x} }{x! } e ^ {- \lambda } \left \{ 1 + \frac{\mu _ {2} - \lambda }{\lambda ^ {2} } \left [ \frac{x ^ {[2]} }{2 } - \lambda x ^ {[1]} + \frac{\lambda ^ {2} }{2 } \right ] \right . +$$

$$+ \left . \frac{\mu _ {3} - 3 \mu _ {2} + 2 \lambda }{\lambda ^ {3} } \left [ \frac{x ^ {[3]} }{6 } - { \frac \lambda {2} } x ^ {[2]} + \frac{\lambda ^ {2} }{2 } x ^ {[1]} - \frac{\lambda ^ {3} }{6 } \right ] \right \} .$$

Here $\mu _ {i}$ are the central moments of the distribution, while $x ^ {[i]} = x( x - 1) \dots ( x - i + 1)$.

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

$$B _ {0} ( x) = \ { \frac{1}{2 \pi } } \int\limits _ {- \pi } ^ { {+ } \pi } e ^ {- itx } \phi ( t) dt.$$

These are convenient for the interpolation between the values $B ( m) = ( n!/m! ( n - m)!) p ^ {m} q ^ {n-m}$ of the general term of the binomial distribution, where

$$\phi ( t) = \ ( pe ^ {it} + q) ^ {n} = \ \sum _ {m = 0 } ^ { n } B ( m) e ^ {itm}$$

is the characteristic function of the binomial distribution. The expansion of $\mathop{\rm ln} \phi ( t)$ in powers of $t$ yields a Gram–Charlier series of type $A$ for $B _ {0} ( x)$, whereas the expansion of $\mathop{\rm ln} \phi ( t)$ in powers of $p$ yields a Gram–Charlier series of type $B$.

#### 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)