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Luis A. Afonso
Posts:
4,758
From:
LIsbon (Portugal)
Registered:
2/16/05


Goodness of fitt : the Population´s "Fingerprints"
Posted:
Oct 10, 2012 12:22 PM


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 ((Ax)/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 90sized 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%.
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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



