hello,Since I'm short of quant skills myself, what would be the appropriate (log) likelihood function of the gram-charlier expansion of a normal standardised variable? i.e. of b(x)*[1+(1/6)*skew*(x^3-3x)+(1/24)*(kurtosis-3)*(x^4-6x^2+3)], where b(x) is standardised binomial density?This would be my guess, but i have a credibility problem in front of myself:log L=c-(n/2)*log*2pi-(n/2)*log*sigma^2-(1/(2*sigma^2))*sum(xi-mju)^2+log*product[1+(1/6)*skew*(xi^3-3xi)+(1/24)*(kurtosis-3)*(xi^4-6xi^2+3)]i can't quite buy my own logic (of just adding the expansion term), so any suggestions?rgdskristjan

Your approach is correct with three qualificiations. The most important is the xi's inside the GC expansion terms should be replaced by (xi - mu)/sigma. Then I assume "mju" is supposed to be "mu" and "log*product. . ." is supposed to be "log(product. . .".You can simplify a bit:(n/2)*log(sigma^2) = n*log(sigma)log(product. . .) = sum(log. . .)For xi far from mu, (xi - mu)^2 will dominate log(4th degree polynomial in xi). But for xi near mu, in terms of number of standard deviation units relative to kurtosis and skewness, the log polynomial will be important.One danger of this approach is you can get negative infinities. Suppose kurtosis is 3 and the skewness term comes out to -1.

Thank you very much! Once i get this coded, i might come back with a similar question about regime switching or mixtures of normals, but I'll be doing some more thorough Hamilton reading first.kristjan