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Topic: Skew, Kurt, Critical Values
Replies: 1   Last Post: May 30, 2013 8:17 PM

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Luis A. Afonso

Posts: 4,613
From: LIsbon (Portugal)
Registered: 2/16/05
Skew, Kurt, Critical Values
Posted: May 30, 2013 10:01 AM
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Skew, Kurt, Critical Values

Just for fun I will evaluate the Skewness and Kurtosis given by Cristiano (thread: Skewness and kurtosis p-values). A very small sample size, as n=10, is a limitation because the PC´s computing slowness. Literature
http://mvpprograms.com/help/mvpstats/distributions/SkewnessCriticalValues
http://mvpprograms.com/help/mvpstats/distributions/KurtosisCriticalValues
The number, 4´000´000, of samples, takes 2 1/2 hors of calculation.
Results: N=10/ 4´000´000

___SKEWNESS*___
_________________Table___
___-1.131__1.133__1.127__0.050, 0.950__
___-1.375__1.378__1.372__0.025, 0.975__
___-1.667__1.675__1.657__0.010, 0.990__
___-1.869__1.876__1.856__0.005, 0.995__
*relative to the two tails separately
___KURTOSIS________Table
___-1.574__2.632___-1.58__2.54__90%
___-1.731__3.447___-1.71__3.44__95%
___-1.886__4.472___-1.88__4.50__99%
In order to check if the RNG is properly working the data 10 * 4000000 concerning the results above, was tested by Lilliefor´s Test
______DATA______
____-3.5___0.00023___3.5___0.99976__
____-3.0___0.00135___3.0___0.99866__
____-2.5___0.00620___2.5___0.99377__
____-2.0___0.02274___2.0___0.97724__
____-1.5___0.06681___1.5___0.93318__
____-1.0___0.15866___1.0___0.84134__
____-0.5___0.30856___0.5___0.69152__
____ 0.0___ 0.50002__
Conclusion: The maximum difference relative to data was dmax=0.191 whereas the critical value, 95% confidence read on Tables amounts to 0.220. So, as expected, there is not sufficient evidence to reject data normality. The overall conclusion is that our Skew, Kurt critical values are in accordance with theory.

Luis A. Afonso
REM "CHECK-0"
CLS
DEFDBL A-Z
PRINT "________check-0_____________________"
PRINT
PRINT
pi = 4 * ATN(1)
INPUT " N= "; n
aw = SQR(n * (n - 1)) / (n - 2)
bw = ((n - 1) * (n + 1)) / ((n - 2) * (n - 3))
cw = -3 * ((n - 1) * (n - 1)) / ((n - 2) * (n - 3))
DIM x(n)
DIM sk(8001), kt(8001)
DIM cc(20)
REM
INPUT " How many "; many

REM
REM
FOR j = 1 TO many
LOCATE 5, 50: PRINT USING "#######"; many - j
RANDOMIZE TIMER
m1 = 0
FOR i = 1 TO n
aa = SQR(-2 * LOG(RND))
x(i) = aa * COS(2 * pi * RND)
x = x(i)
m1 = m1 + x / n
FOR ki = 1 TO 15: REM Normal items check
p = -4 + ki * .5
IF x <= p THEN cc(ki) = cc(ki) + 1 / n
NEXT ki
NEXT i: REM the sample is got
REM
m(2) = 0: m(3) = 0: m(4) = 0
FOR k = 2 TO 4: ki = k
FOR i = 1 TO n: d = x(i) - m1: uu = d ^ ki
m(k) = m(k) + uu / n
NEXT i
NEXT k
REM Skewness (sk) & Kurtosis (kt)
sk = aw * m(3) / (m(2) ^ 1.5)
kt = bw * m(4) / (m(2) * m(2)) + cw
REM
sk = sk + 4: kt = kt + 3
sk = INT(1000 * sk + .5)
kt = INT(1000 * kt + .5)
REM
IF sk > 8000 THEN sk = 8000
IF sk < 0 THEN sk = 0
sk(sk) = sk(sk) + 1
REM
IF kt > 8000 THEN kt = 8000
IF kt < 0 THEN kt = 0
kt(kt) = kt(kt) + 1
REM
uww = INT((10 * j) / many)
IF uww <> (10 * j) / many THEN GOTO 89
REM
COLOR 7
v(0) = .05: v(1) = 1 - v(0)
v(2) = .025: v(3) = 1 - v(2)
v(4) = .01: v(5) = 1 - v(4)
v(6) = .005: v(7) = 1 - v(6)
LOCATE 7, 1
PRINT " SKEWNESS "
FOR w = 0 TO 7
s = 0
FOR t = 0 TO 8000
s = s + sk(t) / j
IF s > v(w) THEN GOTO 15
NEXT t
15 PRINT USING " ##.### "; t / 1000 - 4; v(w);
IF w = 1 OR w = 3 OR w = 5 THEN PRINT
NEXT w: PRINT
PRINT : PRINT " KURTOSIS "
FOR w = 0 TO 5
s = 0
FOR t = 0 TO 8000
s = s + kt(t) / j
IF s > v(w) THEN GOTO 16
NEXT t
16 PRINT USING " ##.### "; t / 1000 - 3; v(w);
IF w = 1 OR w = 3 THEN PRINT
NEXT w
PRINT : PRINT
COLOR 14
FOR p = 1 TO 15
PRINT USING " ##.# #.####### ";
-4 + .5 * p; cc(p) / j;
NEXT p
COLOR 7
89 NEXT j
END
_________________________________
REM "LILI10"
REM
CLS
DEFDBL A-Z
DIM vview(15), ttab(15)
DATA 0.00023,0.00135,0.00620,0.02274,0.06681
DATA 0.15866,0.30856,0.50002,0.69152,0.84134
DATA 0.93318,0.97724,0.99377,0.99866,0.99976
FOR i = 1 TO 15: READ vview(i): NEXT i
DATA 0.00023,0.00135,0.00621,0.02275,0.06681
DATA 0.15866,0.30854,0.50000,0.68146,0.84134
DATA 0.93319,0.97725,0.99379,0.99865,0.99977
FOR i = 1 TO 15: READ ttab(i): NEXT i
REM LILIEFORS TEST
REM max[abs(vview(k-1)-ttab(k),abs(vview(k)-ttab(k)]
vview(0) = 0
FOR k = 1 TO 15
d1 = ABS(vview(k - 1) - ttab(k))
d2 = ABS(vview(k) - ttab(k))
maxx = d1
IF d2 > maxx THEN maxx = d2
IF maxx > supper THEN supper = maxx
NEXT k
LOCATE 7, 20: COLOR 14
PRINT USING " This trial ##.### "; supper
LOCATE 12, 20: COLOR 12
PRINT " 0.220 (95%) Lillieforïs Test"
COLOR 7
END



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