[R] Fitting 4 moments distribution w/ Mixture Gaussian

[Sam R]

Hello, 

I know that Mclust does the fitting on its own but I am trying to implement
an optimization with the aim to generate a the mixture gaussian with the
combine moments as closed as possible to the moment of my return
distribution. 

The objective is to Min Abs((Mean Ret - MeanFit)/Mean Fit) + Abs((Std Ret
-Stdev Fit)/Stdev) + Abs((Sk Ret-Sk fit)/Sk Fit) + Abs((Kurt Ret- Kurt Fit)) 

Taking into account that I fix the weight between the two gaussians at
(0.2;0.8) I implemented the below code in R: 

distance <-function(parameter,x) { 
 u=mean(x) 
 s=sd(x) 
 sk=skewness(x) 
 kurt=kurtosis(x) 
 d1=dnorm(x,parameter[1],parameter[2]) 
 d2=dnorm(x,parameter[3],parameter[4]) 
 dfit=0.2d1+0.8d2 
 ufit=mean(dfit) 
 sdfit=sd(dfit) 
 skfit=skewness(fit) 
 kurtfit=kurtosis(fit) 



abs((u-ufit)/ufit)+abs(s-sdfit)/sdfit)+abs((sk-skfit)/skfit)+abs((kurt-kurtfit)/kurtfit)) 
} 
Parameter<-c(0,0.01,0,0.01)  ' starting point of the optimization 
opp<-optim(parameter,distance,x=conv) 

1/ could anybody tell me whether it is the right approach ? 
2/ should I add some constraint like 
 ufit=0.2*mean(d1)+0.8*mean(d2)... 

thank you very much in advance for your time and help. 

Sam

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