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Topic: Jarque-Bera statistics guessing Normal or Uniform?
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Luis A. Afonso

Posts: 4,617
From: LIsbon (Portugal)
Registered: 2/16/05
Jarque-Bera statistics guessing Normal or Uniform?
Posted: Apr 13, 2013 10:13 AM
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Jarque-Bera statistics guessing Normal or Uniform?


The Classical J.Neyman-E. Pearson NHST scheme to decide if a Null Hypothesis should be rejected can be used in order to choose what the preferential distribution to fit a data-problem is. In fact, allied to systematic simulation concerning the Null, H0, Normal Population, and Alternative Hypotheses, H1, Uniform Population, and the sample response towards a GoF test concerning the Null and an adequate Population parameters we can easily achieve our goal.
Here it is intended, to illustrate the progressive ability to discriminate Uniformity from Normality by the Jarque-Bera (ALM) algorithm in conjunction with the Skewness and Excess Kurtosis indexes when sample sizes are let to vary from 50 to 100 (400?000 samples each time). No specific data-sample is used simulated instead does. So, the test will inform how sizes became untenable towards Normality against Uniformity when we compare the observed triple outputs/responses JB, S, and k, codified by [000] to [111], 1 indicating significant result, 0 not.

The proposed Method

Firstly of all the 5% critical values:
____________JBcrit(5%)____________1%______
____50______6.55__6.55_______16.63__16.62__
____60______6.41__6.41_______16.23__16.19__
____70______6.41__6.40_______15.68__15.68__
____80______6.38__6.39_______15.31__15.32__
____90______6.34__6.33_______14.96__14.92__
___100______6.36__6.31_______14.71__14.73__

and
________________5%_____________1%_______
____________Ucrit__Vcrit___
___50_______3.92___3.44______7.39___11.02__
___60_______3.91___3.44______7.30___10.81__
___70_______3.90___3.45______7.25___10.52__
___80_______3.91___3.47______7.19___10.44__
___90_______3.90___3.48______7.12___10.15__
__100_______3.90___3.52______7.03____9.99__

Finally the *ouputs* using JBcrit, Ucrit,Vcrit (5% level)
After carefully consideering vantages and drawbacks to obtain more precise critical values I opt to choose the following ones, taking into account that semi-quantitative goal of the present account is not extremely demanding.

JBcrit= 6.55, 6.41, 6.40, 6.39, 6.35, 6.35
Ucrit = 3.91______________________
Vcrit = 3.44, 3.44, 3.44, 3.47, 3.47, 3.52
______________________________________

Table__ Jarque-Bera test outputs, ALM

__Normal (Gaussian)
__Uniform

__[000]__[001]__[010]___[011]__[100]__[101]__[110]__[111]_

__n=50____
__0.921__0.012__0.017__0.000__0.000__0.016__0.011__0.022_
__0.629__0.369__0.002__0.000__0.000__0.000__0.000__0.000_

__n=60____
__0.919__0.013__0.018__0.000__0.001__0.018__0.012__0.020_
__0.406__0.585__0.002__0.000__0.000__0.007__0.000__0.000_

__n=70____
__0.918__0.013__0.019__0.000__0.001__0.018__0.013__0.019_
__0.228__0.722__0.001__0.000__0.000__0.048__0.001__0.000_

__n=80____
__0.917__0.014__0.019__0.000__0.001__0.019__0.013__0.018_
__0.119__0.721__0.001__0.000__0.001__0.157__0.001__0.000_

__n=90____
__0.916__0.015__0.019__0.000__0.001__0.019__0.013__0.019_
__0.055__0.601__0.000__0.000__0.001__0.341__0.001__0.000_

Supposing that you had found [001] or [101], which is, of course, very likely to occur with H1 true, then you decide to reject H0 because p= 0.601+ 0.341 = 94.2% if. The decision has the risk 0.015+ 0.019 = 3.4% to fail.
On contrary if you got [000] the H1 rejection has only 5.5% probability, and to retain H0, p= 91.6%, is advisable.

__n=100____
__0.915__0.015__0.020__0.000__0.001¬__0.019__0.014__0.016_
__0.025__0.436__0.000__0.000__0.001__0.537__0.001__0.000_
__p = 0.436 + 0.537 = 97.3% if H1=true and get [001] or [101]. Contrasting with p([001]or[101] | H0 true) = 3.4%.


Luis A. Afonso



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