< Probability Distribution List < Gram-Charlier Distribution
The Gram-Charlier distribution is a way to explicitly model departure from normality by using a series expansion around a normal distribution; A series expansion is a way to represent a function as a sum of powers in one variable, or by a sum of powers of another function.
The Gram-Charlier is more flexible than a normal distribution because it directly introduces a distribution’s kurtosis and skew as unknown parameters. The normal distribution has a kurtosis of 3 and a skewness of 0. The Gram-Charlier distribution, on the other hand, can have any values for kurtosis and skewness. This means that the Gram-Charlier can be used to model a wider range of distributions than the normal distribution.
One practical application of the Gram-Charlier distribution is to model stock returns in options pricing. It can also be used to model the distribution of insurance claims and model the distribution of risks in risk management.
Types of Gram-Charlier Distribution
The two-term Gram-Charlier distribution is defined by  as
Type A is defined by :
Where Hei is a Hermite polynomial defined as
The first six of which are:
- He0(z) = 1
- He1(z) = z
- He2(z) = z2 – 1
- He3(z) = z3 – 3z
- He4(z) = z4 – 6z2 + 3
- He5(z) = z5 – 10x3 + 15z
- He6(z) = z6 – 15z4 + 45z2 – 15
Quensel  presented a logarithmic Gram-Charlier distribution, where log X has a Gram-Charlier distribution.
Drawbacks of the Gram-Charlier Distribution
The Gram-Charlier distribution does have a couple of drawbacks. As it involves polynomial approximations, it can result in negative parameter values under certain conditions. In addition, there isn’t an easy and analytic characterization of a density which will take on only positive values . Gallant and Tauchen  suggested simply squaring the polynomial part of the series. However, that results in losing interpretation of parameters as moments. that A combined model may also be used to exclude negative values; where the range of random variables are small, the PDF is described by a truncated Gram-Charlier distribution. Outside of that area, the model is a normal distribution .
 Haight, F. (1958). Index to the Distributions of Mathematical Statistics. National Bureau of Standards Report.
 Jondeau, E. & Rockinger, M. (1999). ESTIMATING GRAM-CHARLIER EXPANSIONS WITH POSITIVITY CONSTRAINTS. Les Notes d’Études et de Recherche. Bank of France. Online: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.867.1681&rep=rep1&type=pdf
 Quensel, C.-E. (1945). Studies of the logarithmic normal curve, Skandinavisk Aktuarietidskrift, 28, 141-153.
 Gallant and Tauchen (1989). Seminonparametric Estimation of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications. Econometrica
Vol. 57, No. 5 (Sep., 1989), pp. 1091-1120 (30 pages). Published By: The Econometric Society
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