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Topic: Jarque-Bera statistics guessing Normal or Uniform?
Replies: 3   Last Post: Apr 18, 2013 6:36 AM

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Luis A. Afonso

Posts: 4,615
From: LIsbon (Portugal)
Registered: 2/16/05
Re: Jarque-Bera statistics guessing Normal or Uniform?
Posted: Apr 17, 2013 5:34 PM
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< Fingerprints > II
Do consider the triplet G, U, V where G is the Geary Normality test, U and V the ALM skewness index, and Excess kurtosis. We start to find the 5% alpha, 1%, ( right tail) Critical Values relative to the former. They are:
_____zw quantiles
_______________5%_________1%_____
____50________1.62________2.46_____
____60________1.62________2.45_____
____70________1.63________2.44_____
____80________1.63________2.44_____
____90________1.64________2.43_____
___100________1.64________2.43_____

____sigma= SQR[sum (Xi - m)^2/ n)]
____tau = sum [ABS (Xi) / n]
____omega= 13.29 * LOG(sigma/ tau)
____zw = sqrt (n+2) * [(omega- 3) /3.54]

Bibliography (net)
D.G.Bonnet, E.Seier - A test of normality with high uniform power, (Computational Statistics & Data Analysis 40 (2002) 435-445.
__________
For n=100, the CV (5%) are respectively 1.64, 3.91, 3.52 (the two later ones found previously at other post).

400´000 samples form N(0,1):100 were put under test. Uniform samples, U=RND and U´=6*(RND- 0.5, as well.

Results
___________N(0,1)________U_________U´____
[000]_______ 0.915_______0.026______0.026__
[001]_______ 0.035_______0.972______0.973__
[010]_______ 0.034_______0.001______0.001__
[011]_______ 0.016________________________

An important feature, indispensable in view its usefulness, is that, whatever U or U´ , the frequencies do not change (parametrically independence).
In conclusion: normal data is distinguishable when the test
____H0: normal, Ha=Uniform
The Type I error: to reject wrongly normal data is 3.5%.
By the other hand, the Type II error, *accepting* U with normal data is 2.6% (Power=97.4%). Remember: Power is the probability not to make a type II error).

Luis A. Afonso



REM "U-PUPOSE"
CLS
COLOR 13: PRINT : PRINT " < U-PUPOSE > "
PRINT " [Geary, U, V ] ALM , uniform "
DEFDBL A-Z
pi = 4 * ATN(1)
RANDOMIZE TIMER
COLOR 14
INPUT " Size (<=2000 ) "; n
INPUT " Gcrit, Ucrit, Vcrit "; Gcrit, Ucrit, Vcrit
INPUT " How many "; many
DIM x(n), U1(8001), V1(8001)
c1 = 6 * (n - 2) / ((n + 1) * (n + 3))
c2 = 3 * (n - 1) / (n + 1)
c3 = 24 * n * (n - 2) * (n - 3) / ((n + 1) ^ 2 * (n + 3) * (n + 5))
REM
COLOR 7
FOR j = 1 TO many
LOCATE 7, 50: PRINT USING "########"; many - j
m = 0
FOR i = 1 TO n
x(i) = 6 * (RND - .5)
m = m + x(i) / n
NEXT i
REM
m2 = 0: m3 = 0: m4 = 0: absx = 0
FOR i2 = 1 TO n: d = x(i2) - m
m2 = m2 + d * d / n
m3 = m3 + d * d * d / n
m4 = m4 + d * d * d * d / n
absx = absx + ABS(d)
NEXT i2
s = m3 / (m2 ^ 1.5): U = s * s / c1
kk = m4 / (m2 * m2) - c2: V = kk * kk / c3
sigma = SQR(m2 / n)
tau = absx / n
omega = 13.29 * LOG(sigma / tau)
zww = SQR(n + 2) * ((omega - 3) / 3.54)
COLOR 7
REM
i1 = 0
IF zww > Gcrit THEN i1 = 1
i2 = 0
IF U > Ucrit THEN i2 = 1
i3 = 0
IF V > Vcrit THEN i3 = 1
REM
W(i1, i2, i3) = W(i1, i2, i3) + 1
REM
NEXT j
PRINT : PRINT : PRINT
FOR i1 = 0 TO 1
FOR i2 = 0 TO 1
FOR i3 = 0 TO 1
PRINT " ";
PRINT USING "[# # #] "; i1; i2; i3;
PRINT USING "#.###"; W(i1, i2, i3) / many
NEXT i3
NEXT i2
NEXT i1
REM
END



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