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Luis A. Afonso
Posts:
4,275
From:
LIsbon (Portugal)
Registered:
2/16/05
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Goodness of fitt : the Population´s "Fingerprints"
Posted:
Oct 10, 2012 12:22 PM
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The Population?s ?Fingerprints?
Here the ?drama? is played by two characters: The team of interval?s tests, with their probabilities to ?capture? the test, no significance, noted 0, or outside, noted 1, by the other hand the Population i.i.d. random samples under test. Therefore, for example, the output/result symbol [011] denotes that the first test is inside and the second and third is no significant. The interval?s scorers, here 3, could be as large as we want, the outputs have an unlimited number of symbols as [0100?10]. This shows that the second score is significant (outside the respective interval), the first, third and fourth no significant.
Given an observed output we can, by Monte Carlo simulating, to evaluate how likely it is given a proposed Population.
We only intend to give an example based on the JB test and the Skewness and Excess Kurtosis parameters for the Populations Normal, Uniform, Gambel (0,1) and (1,2), family CDF = exp(-exp ((A-x)/B)), which inverted give directly x = A - B*log (-log (CDF)), used to simulate i.i.d. samples.
Analysis the Table below we can, for example, to state that a Uniform 90-sized sample ?cannot? show [000] or [110], probability 0.011 for this issue. Therefore if we have the chance to observe [000] or [110] we immediately exclude Uniformity, of course. On contrary the Normal Distribution is quite likely 46.5 + 20.3= 66.8%.
_____________________
Table: Critical Values: Skewness, S, and Excess Kurtosis, k, 2.5% significance level, JB test 5%, for sample sizes 60 (10) 100, normal data, and U(-3, 3) obtained from 1 million samples each, JB, 4 million.
_Normal Data
_______ JB(.975)_____S(2.5%)________k(2.5%)_____
__N___
__60_____7.75____[-0.696, 0.696]___[-0.949, 1.622]__
__70_____7.78____[-0.645, 0.645]___[-0.902, 1.521]__
__80_____7.86____[-0.604, 0.604]___[-0.860, 1.419]__
__90_____7.95____[-0.572, 0.572]___[-0.823, 1.362]__
_100_____7.95____[-0.543, 0.543]___[-0.792, 1.291]__
When we test normality , using the intervals above, and data are Normal/Uniforme/Gumbel(0,1)/Gumbel(1,2) we obtain the following results (0 denoting inside, 1 outside):
OUPUT PERFILES
_size=_____60____ 70_____80____90____100__
__[000]__ 0.657__0.591__0.524__0.465__0.405__N
_________0.108__0.053__0.025__0.011__0.005__U
_________0.302__0.224__0.164__0.121__0.086__G(0,1)
___________0_____0_____ 0______0_____0____G(1,2)
__[001]__ 0.029__0.028__0.029__0.029__0.028__
_________0.892__0.947__0.974__0.980__0.943__
_________0.004__0.003__0.002__0.002__0.001__
___________0_____0_____ 0______0_____0____
__[010]__0.126__0.155__0.182__0.203__0.214__
___________0_____0_____0______0______0____
________0.125__0.125__0.119__0.107__0.093__
___________0_____0_____0______0______0____
__[011]____ 0_____0_____0______0______0____
___________0_____0_____0______0______0____
___________0_____0_____0______0______0____
___________0_____0_____0______0______0____
__[100]__0.001__0.001___ 0______0______0____
___________0_____0_____0______0______0____
________ 0.001__0.001__0.001___ 0______0____
___________0_____0_____0______0______0____
__[101]____ 0_____0_____0______0______0____
___________0_____0___0.001__0.010__0.052____
___________0_____0_____0______0______0____
________0.306__0.229__0.170__0.123__0.089___
__[110]_ 0.132__0.167__0.201__0.237__0.282__
___________0_____0_____0______0______0___
________0.265__0.304__0.332_ 0.355__0.368___
___________0_____0_____0______0______0___
__[111]_ 0.054__0.059__0.063__0.065__0.070__
___________0_____0_____0______0______0____
________0.303__0.342__0.382_ 0.415__0.450___
________0.694__0.771__0.830_ 0.877__0.911_
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It?s irrelevant the parameters exact values for these tables construction. In fact, the ways S and k are defined are automatically ?standardized? in what concerns the normal Population mean value and standard deviation. Otherwise one was compelled to adjust the estimations obtained from the real sample to the estimated ones by Monte Carlo, and the Table would loss generality.
Luis A. Afonso
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