I recommend either
* doing the theoretical computations; not difficult using cumulants (below).
* use saddlepoint approximations, if the cumulant generating function
is tractable. This is much more accurate for tail approximations than
Edgeworth or Gram-Charlier expansions based on moments.
Let M_j represent the centered j'th moment of X, and
N_j the centered j'th moment of S = sum_{i=1}^n X_i
Then if the X's are i.i.d.
N_2 = n M_2
N_3 = n M_3
N_4 = n M_4 + 3 (n^2-n) M_2^2
>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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