moments of sums

[Kamil Toth <kamiltoth@yahoo.com>]

Dear S-Friends:

 

Please, help to resolve a seemingly simple question.

 

I need to compute a higher (say, up to forth) moments of sum(x)  

where x[1:N] are non-normal iid. The structure is such that the conversion to asymptotic normal limit,  if exists, is slow, therefore the central limit theorem is not working for reasonably moderate N, say 100. Because it is difficult to perform such a computation analytically, I have devised the following idea.

I know the first four theoretical moments mu1,mu2,mu3,mu4 of the individual

"x". I can reasonable expect that the fourth moment MU4 of sum(x)

is somehow expressed through the theoretical moments mu1,mu2,mu3,mu4.

And based on the considerations of dimension I try to fit:

 

MU4=a*mu4+b*mu3*mu1+c*mu2*mu1^2+d*mu1^4

 

Then I try to find coefficients a,b,c,d, using "lm" in Monte Carlo simulation

 

below is the example of the dataframe I use (first 9 MC iterations)

 

 

 MU1 MU2   MU3   MU4      mu1 mu2  mu3   mu4   
4.90 29.10 202.30 1609.26 0.5 0.75 1.375 3.0625
5.31 33.53 243.69 1993.49 0.5 0.75 1.375 3.0625
5.08 31.26 220.90 1724.82 0.5 0.75 1.375 3.0625
4.96 28.06 174.58 1169.14 0.5 0.75 1.375 3.0625
4.82 27.54 173.24 1165.74 0.5 0.75 1.375 3.0625
4.68 27.34 182.64 1340.86 0.5 0.75 1.375 3.0625
4.83 29.19 214.05 1895.79 0.5 0.75 1.375 3.0625
5.21 31.97 220.49 1662.77 0.5 0.75 1.375 3.0625
4.74 28.08 193.68 1491.00 0.5 0.75 1.375 3.0625

 

the standard lm procedure terminates with error message : singular matrix.

 

Does anybody have a clue what is wrong in my approach?

If so, how should I proceed? Any ideas or references are highly appreciated

 

Thanks in advance

 

Kamil Toth

 

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