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Topic: Skewness (Gumbel data) comparison by simulation
Replies: 1   Last Post: Apr 2, 2014 7:16 AM

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 Luis A. Afonso Posts: 4,758 From: LIsbon (Portugal) Registered: 2/16/05
Skewness (Gumbel data) comparison by simulation
Posted: Apr 1, 2014 7:36 PM

Introduction
Even a Distribution with zero skewness (symmetrical) the random samples shows values that are somewhat smaller or larger than this value, the imprecision going less and less when the sample size increases. We intend to compare the observed value with those the different Gumbel (A, B) Distributions do show, A and B varying at given intervals, in order that we assure we can reject that the sample could be drawn from them. More directly we will find by random simulation what sample size should be as large that only a small proportion of samples skewness, the current 5%, are less than that a list of values the skewness is characterized. The sample sizes were 150 (50) 350 , 0 <= A, B >= 1. We found:
Prob. ( Observ. skewness | Gumbel(A, B) skewness) < 0.05
Results
__n= 150___<= -1.75__n= 200___<= -1.69__n=250___ <= -1.65
__n=300___ <=-1.61
The number of samples involved was 4000, therefore we do not reject H0: same skewness than Gumbel (A, B) since the number of values does not exceed 200 approx.
A detailed example (2 attempts)
__n=150
______________bound= -1.75___________-1.73____
__(0.00 , 0.00)__0.045__0.045________0.051__0.055
__(_____, 0.50)__0.049__0.049________0.050__0.049
__(_____, 1.00)__0.044__0.046________0.048__0.050
__(0.50 , 0.00)__0.049__0.047________0.050__0.053
__(_____, 0.50)__0.046__0.045________0.050__0.049
__(_____, 1.00)__0.046__0.045________0.054__0.050
__(1.00 , 0.00)__0.050__0.044________0.052__0.056
__(_____, 0.50)__0.051__0.046________0.049__0.050
__(_____, 1.00)__0.053__0.052________0.051__0.050

Luis A. Afonso

REM "ANTIGO"
CLS
PRINT " <ANTIGO> (5TIMES-G) "
DEFDBL A-Z
RANDOMIZE TIMER
PRINT " 5 Bounds ";
FOR gh = 1 TO 5
bnd(gh) = -1.77 + .02 * gh : REM adapted to the case
NEXT gh
INPUT " size = "; size
REM
DIM X(size)
REM
all = 4000
REM
FOR inspec = 1 TO 5
COLOR inspec + 1
PRINT USING " under test 95% fractil ###.## "; bnd(inspec)
REM
FOR u = 0 TO 1 STEP .5: A = u
FOR v = 0 TO 1 STEP .5: B = v
REM
outt = 0
FOR j = 1 TO all
mi = 0
FOR i = 1 TO size
22 rn = RND
IF rn < .000001 THEN GOTO 22
X(i) = A - B + LOG(-LOG(rn))
mi = mi + X(i) / size
NEXT i
m2 = 0: m3 = 0
FOR vi = 1 TO size: d = X(vi) - mi
m2 = m2 + d * d / size
m3 = m3 + d * d * d / size
NEXT vi
sk = m3 / (m2 ^ 1.5)
IF sk < bnd(inspec) THEN outt = outt + 1
NEXT j
IF outt > .05 * all THEN GOTO 25
PRINT USING "(##.## ##.##) #.### "; A; B; outt / all;
REM
25 outt = 0
NEXT v
REM
NEXT u
PRINT
NEXT inspec
REM
END

Date Subject Author
4/1/14 Luis A. Afonso
4/2/14 Luis A. Afonso