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Luis A. Afonso

Posts: 4,277
From: LIsbon (Portugal)
Registered: 2/16/05

<Grumbling> about Jarque-Bera Test: Do you would give a hint to mend it?
Posted: Oct 4, 2012 4:48 PM

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<Grumbling> about Jarque-Bera Test: Do you would give a hint to mend it?

The difficulty to approve the test is that to simply using addition, or whatever function, as Jarque and Bera chose, to relate two-parameter?s estimates is, in fact, an erroneous way to check they are both conformable with a normal law.

So, in this context,
___1___A worthy feature is that the test is completely independent from the proposed Test Statistics.
___2___The critical values for each result are found by Monte Carlo simulation as it was improved in what concerns J-B test.
___3___The present method can be extended to a Goodness of Fit test of whatever number of simultaneous parameters: results are such (0, 1, 1, ?, 0).
_______

Example

An observed size 10 data-vector (after ordering) is:
107, 110, 111, 112, 113, 114, 114, 116, 117, 120.
Mean = 113.4____St.dev. = 3.71777
Skewness = 0.07767
Excess Kurtosis = -0.44123

By simulation (1 million 10 size samples) we got the confidence intervals for each parameter:

Skew:
_____[-1.73 ; 1.73 ]____(0.0025; 0.9775)
_____[-1.35 ; 1.35 ]____(0.0125; 0.9875)
Exc. Kurtosis:
_____[-1.70 ; 2.76 ]
_____[-1.59 ; 1.85 ]

This shows that the Skewness Coefficient is Symmetrical about 0, but contrarily the Excess Kurtosis has a long right tail. For our sample we fail to reject H0, there is not sufficient evidence that is not normal, even if alphaBonferroni= alpha/2; so, for each tail = 0.05/4 = 0.0125, as shown (second lines) above.

We think this is better than J-B, the only not indisputable point is the criterion to use the critical Bonferroni´s composite alpha equal to alpha/2, the remaining of the method, based on simulating data, is model-exact.
A related, but rather different problem is that of the lack of selectivity, particularly for small samples: one is ?outrageously? on risk to conclude wrongly by normality when is not the case (type II error).
The really drama of Statistical Inference is that we are always dealing with necessary not sufficient mathematical ?tools?.

Luis A. Afonso