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Topic: A pertinent/impertinent question. . .
Replies: 8   Last Post: Nov 13, 2013 4:02 PM

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Luis A. Afonso

Posts: 4,613
From: LIsbon (Portugal)
Registered: 2/16/05
Re: A pertinent/impertinent question. . .
Posted: Nov 3, 2013 7:54 AM
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Towards asymptotic distribution functions

Since, as was yet found in literature, that either U and V,
_____U= n/6* [m3/(m2^1.5)]^2_____(a1)
_____V= n/24*[m4/(m2^2 - 3]^2____(a2)
JB=U+V, does not follow each one, for moderated size samples, a chi squared 1 degree of freedom (df) Distribution, it was thought that to use exact quantiles of JB statistics vs. the sizes, n, would solve the problem rigorously.
However, clearly, this is not the case: each estimation formula have its own speed to became stable and equal each other.


_A)__ similarity asymptotic
___n=4000/40´000____(2 experiments)
The formulas (a1),(a2) above concerns the LM version[1].


___U_____V_______U_____V____q__
__0.30___0.25_____0.26___0.26__(.4)
__0.47___0.42_____0.44___0.42__(.5)
__0.72___0.68_____0.68___0.67__(.6)
__1.08___1.03_____1.03___1.03__(.7)
__1.63___1.60_____1.59___1.59__(.8)
__2.70___2.66_____2.66___2.62__(.9)

Note: the quantiles (from .4 to .9) seems not to differ for the two r.v.. Furthermore should be compared with those from the Chi-square: 0.275, 0.455, 0.708, 1.074, 1.642, 2.706.
It is good practice do not use the JB test for samples shorter than 4000. Otherwise the contributions U and V to the total are quite unbalanced resulting that we are testing the sample problem not fairly regarding both parameters.

B) In what concerns the ALM procedure [1] it was not expected differences for large samples because it not modifies the estimates.

Bibliography

[1] - arXiv:math/0509423v1 [math.ST] 19 Sep 2005.


Luis A. Afonso



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