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Topic: Testing normality by Skewness and Kurtosis: a new focusing
Replies: 1   Last Post: Oct 28, 2013 7:46 AM

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Luis A. Afonso

Posts: 4,613
From: LIsbon (Portugal)
Registered: 2/16/05
Testing normality by Skewness and Kurtosis: a new focusing
Posted: Oct 24, 2013 12:51 PM
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Testing normality by Skewness and Kurtosis: a new focusing



___0___Preliminaries. Aim.

We must agree that the so-called Jarque-Bera Test is wrong: it fails its aim of checking normality through two-parameters estimation, the Skewness S and (Excess) Kurtosis, k. The reason being elementary, surprisingly, I did not find any objection at this point. In fact how to perform the parameter´s sum, in order to obtain the test statistics, we immediately lose the individual capacity they are, or not, well fitted to normality, the Null Hypothesis. The only claim we found concerns the weak power the test shows but the main reason of such feature was not disclosed.
The problem can be solved keeping the two parameters at work, but estimating the individual worth against the Null Hypothesis, in order not to allow a trade-off between them, as the additive procedure does. The Jarque-Bera falsity can be expressed easily: Suppose that S and k are such that the parameters sum U´´ + V´´ are exactly equal to the J-B critical value, noted JBcrit. Then we conclude that, whatever x, we have always (U``+ x) + (V``- x) = JBcrit, a not rejection condition, no matter the latter parameters oddness in what concerns normality.

___1___The new paradigm

The procedure begins to obtain, by simulation, a not-too-tight net of S, k critical values, by simulation, individually, for example regarding alpha=0.095(0.005)0.055 and can be supplied by sufficiently detailed Tables, therefore not using <Sk-crit> routine, below . The second stage consists in to evaluate how each pair S, k, is situated, inside or outside, the Confidence Intervals, given by the above bonds, and chose the alpha which contains, approximately, 95% of the simulated pairs. Because these bounds are intrinsically a propriety of all normal samples we can consider with the size chosen, of course.

Follows an example of how to find out the optimal D and k 95% confidence intervals.

Example, size 10 samples:

__n=10___________S__________k_______________p____
Alpha = 0.095___+/-.65____-1.01,1.29__________0.932
??= 0.090___+/-.67____-1.02,1.34__________0.937
??= 0.085___+/-.69____-1.04,1.39__________0.938
??= 0.080___+/-.70 ___ -1.05,1.44__________0.944
??= 0.075___+/-.72 ___ -1.07,1.50__________0.944
??= 0.070___+/-.74 ___ -1.08,1.56__________0.9495
??= 0.065___+/-.76 ___ -1.10,1.63__________0.9544
??= 0.060___+/-.79 ___ -1.11,1.70__________0.957
??= 0.055___+/-.81 ___ -1.13,1.78__________0.961
??
The acceptance intervals are [+/- 0.742] and [-1.081, 1.560] for all n=10 normal samples, see <What>. To real world samples with both S and k estimated parameters inside are likely normal. Note that classical Statistics NHST are dealing only with necessary results, never sufficient ones.

Luis A. Afonso

REM "SK-crit"
CLS
PRINT : PRINT "_______Sk-crit (critical values)_______"
DEFDBL A-Z
RANDOMIZE TIMER
pi = 4 * ATN(1)
INPUT " size = "; n
INPUT " all = "; all
REM
cs = 1 / (n - 2) * SQR(n * (n - 1))
cn = ((n - 1) * (n + 1)) / ((n - 2) * (n - 3))
cnn = ((n - 1) * (n - 1)) / ((n - 2) * (n - 3))
DIM scv(8001), kcv(8000)
DIM X(n)
FOR j = 1 TO all
REM
LOCATE 4, 50: PRINT USING "########"; all - j
m = 0: m(2) = 0: m(3) = 0: m(4) = 0
FOR i = 1 TO n
aa = SQR(-2 * LOG(RND))
X(i) = aa * COS(2 * pi * RND)
m = m + X(i) / n
NEXT i
FOR k = 2 TO 4
FOR i = 1 TO n: d = X(i) - m
m(k) = m(k) + d ^ k / n
NEXT i: NEXT k
S0 = m(3) / ((m(2) ^ 1.5))
s = cs * S0
k = cn * m(4) / (m(2) * m(2)) - 3 * cnn
suv = INT(1000 * (s + 4) + .5)
IF suv > 8000 THEN suv = 8000
kuv = INT(1000 * (k + 4) + .5)
IF kuv > 8000 THEN kuv = 8000
REM
scv(suv) = scv(suv) + 1 / all
kcv(kuv) = kcv(kuv) + 1 / all
NEXT j
REM PRINT " ";
REM PRINT " skewness excess kurtosis "
REM
c(1, 1) = .095: c(1, 2) = 1 - c(1, 1)
c(2, 1) = .09: c(2, 2) = 1 - c(2, 1)
c(3, 1) = .085: c(3, 2) = 1 - c(3, 1)
c(4, 1) = .08: c(4, 2) = 1 - c(4, 1)
c(5, 1) = .075: c(5, 2) = 1 - c(5, 1)
c(6, 1) = .07: c(6, 2) = 1 - c(6, 1)
c(7, 1) = .065: c(7, 2) = 1 - c(7, 1)
c(8, 1) = .06: c(8, 2) = 1 - c(8, 1)
c(9, 1) = .055: c(9, 2) = 1 - c(9, 1)
REM
FOR ju = 1 TO 9
REM
COLOR ju + 1
FOR kk = 1 TO 2
s = 0
FOR t = 0 TO 8000
s = s + scv(t)
IF s > c(ju, kk) THEN GOTO 1
NEXT t
1 LOCATE 3 + 2 * ju + kk, 10
a(ju, kk) = t / 1000 - 4
PRINT USING "##.### .### "; a(ju, kk); s;
NEXT kk
REM
FOR kk = 1 TO 2
s = 0
FOR t = 0 TO 8000
s = s + kcv(t)
IF s > c(ju, kk) THEN GOTO 2
NEXT t
2 LOCATE 3 + 2 * ju + kk, 30
PRINT USING "##.### .### "; t / 1000 - 4; s;
NEXT kk
NEXT ju
COLOR 7
END

REM "WHAT"
CLS
PRINT : PRINT "_______WHAT_______"
DEFDBL A-Z
RANDOMIZE TIMER
pi = 4 * ATN(1)
INPUT " size = "; n
INPUT " CI S "; bound1, bound2
INPUT " k "; bounty1, bounty2
INPUT " all = "; all
REM
cs = 1 / (n - 2) * SQR(n * (n - 1))
cn = ((n - 1) * (n + 1)) / ((n - 2) * (n - 3))
cnn = ((n - 1) * (n - 1)) / ((n - 2) * (n - 3))
DIM scv(8001), kcv(8000)
DIM X(n)
FOR j = 1 TO all
REM
LOCATE 10, 50: PRINT USING "########"; all - j
m = 0: m(2) = 0: m(3) = 0: m(4) = 0
FOR i = 1 TO n
aa = SQR(-2 * LOG(RND))
X(i) = aa * COS(2 * pi * RND)
m = m + X(i) / n
NEXT i
FOR k = 2 TO 4
FOR i = 1 TO n: d = X(i) - m
m(k) = m(k) + d ^ k / n
NEXT i: NEXT k
S0 = m(3) / ((m(2) ^ 1.5))
s = cs * S0
k = cn * m(4) / (m(2) * m(2)) - 3 * cnn
pinn = 0
IF s < bound1 OR s > bound2 THEN pinn = pinn + 1
IF k < bounty1 OR k > bounty2 THEN pinn = pinn + 1
IF pinn = 2 THEN bothout = bothout + 1 / all
NEXT j
PRINT USING "inside = #.####"; 1 - bothout
END



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