Luis A. Afonso
Posts:
4,699
From:
LIsbon (Portugal)
Registered:
2/16/05


Re: JarqueBera statistics guessing Normal or Uniform?
Posted:
Apr 17, 2013 5:34 PM


< 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) 435445. __________ 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 "UPUPOSE" CLS COLOR 13: PRINT : PRINT " < UPUPOSE > " PRINT " [Geary, U, V ] ALM , uniform " DEFDBL AZ 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

