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Topic: An experiment . . . at random in intra-Permutations
Replies: 5   Last Post: Apr 2, 2013 7:53 PM

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Luis A. Afonso

Posts: 4,526
From: LIsbon (Portugal)
Registered: 2/16/05
Re: An experiment . . . at random in intra-Permutations
Posted: Mar 6, 2013 9:50 AM
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__0__Introduction

After several unsuccessful days with failed attempts I got an algorithm able to achieve the goal to find a Confidence Interval for the difference of two sample variances using the general intra-permutation no-parametric method where the exhaustive no-replacement draws of the two source sample items are performed in order to construct the pseudo-samples.
__1__The Procedure
Let X be a real-world sample, size nX, mean mX, whose items we randomly draw in order to form a pseudo-sample X´. After to to get it we have the current variance j given by:
sdX(j)= (1<=i<=nX) Sum (X´(g) - mX)^2*Wx(i)/(nX-1)
where Wx(i) = i/(1+2+?+n), index showing the order the item is drawn: g=INT(nX*RND)+1. Once an item is chosen it is made unable to be redrawn in the current sample.
Same operations for the sample Y, leading to d(j) as the generic member d(j) = sdX(j) - sdY(j).

__2__Illustrative example: the spider´s data (size 30) presented at my Feb 24, 2013 11:11 AM post.(see program below)

________Female _______________¬¬¬¬__Male__
____sample mean= 8.127__________5.917___
________std. dev.= 1.134__________0.663___
______ Skewness = 1.0269 ________ 1.0181__
___Exc. Kurtosis =-1.9287________-1.9512__

A 95% Confidence interval 20´000 iterations, provided [0.528, 1.165] for the difference of variances. So, one can conclude that at least the difference is 0.528, with 5% significance. Repeating with 1 million it was got an interval [0.525, 1.164]. As expected these intervals are centered at the source difference variances 0.846 approx.
Luis A. Afonso
REM "varSPID"
REM
CLS
PRINT : PRINT "______VARSPID______";
PRINT " Spiderïs intra-Permutation___variances"
pi = 4 * ATN(1)
DIM X(30), Y(30), xx(30), YY(30)
DIM W(9000), wx(30), wy(30)
INPUT " many= "; many
REM male
DATA 4.70,4.70,4.80,5.20,5.20,5.40,5.50,5.65,5.65,5.70
DATA 5.75,5.75,5.75,5.80,5.85,5.85,5.90,5.95,5.95,6.10
DATA 6.20,6.20,6.35,6.35,6.45,6.55,6.80,6.95,7.00,7.50
REM female
DATA 5.90,6.10,6.30,6.60,7.00,7.05,7.05,7.50,7.55,7.55
DATA 7.80,7.95,8.00,8.00,8.10,8.25,8.30,8.30,8.35,8.45
DATA 8.70,8.75,9.00,9.10,9.30,9.50,9.60,9.80,9.95,10.00
REM
n = 30
FOR i = 1 TO n: READ Y(i): YY(i) = Y(i)
msY = msY + Y(i) / n: YY = YY + Y(i) * Y(i): NEXT i
FOR i = 1 TO n: READ X(i): xx(i) = X(i)
msX = msX + X(i) / n: xx = xx + X(i) * X(i): NEXT i
varX = (xx - n * msX * msX) / (n - 1)
varY = (YY - n * msY * msY) / (n - 1)
PRINT "*** var female - var male ---> ";
PRINT USING "#.### "; varX - varY
REM
FOR t = 1 TO 30: si = si + t: NEXT t
FOR t = 1 TO 30: W(t) = t / si: NEXT t
REM
REM
PRINT : COLOR 14
FOR tur = 1 TO many
RANDOMIZE TIMER
LOCATE 5, 55
PRINT USING "#########"; many - tur
FOR i = 1 TO n: xx(i) = X(i): NEXT i
FOR i = 1 TO n: YY(i) = Y(i): NEXT i
REM
gxx = 0
FOR t = 1 TO n
1 g = INT(n * RND) + 1
IF xx(g) = 123456 THEN GOTO 1
gxx = gxx + W(t) * (xx(g) - msX) * (xx(g) - msX)
xx(g) = 123456
NEXT t
gyy = 0
FOR t = 1 TO n
2 g = INT(n * RND) + 1
IF YY(g) = 123456 THEN GOTO 2
gyy = gyy + W(t) * (YY(g) - msY) * (YY(g) - msY)
YY(g) = 123456
NEXT t
varX = n * gxx / (n - 1): REM gxx is a mean value
varY = n * gyy / (n - 1)
d = varX - varY
d1 = INT(1000 * d + .5)
REM
REM
REM
W(d1) = W(d1) + 1
NEXT tur
REM
COLOR 14
PRINT " Confidence Interval 95% "
u(1) = .025: u(2) = 1 - u(1)
FOR uu = 1 TO 2
sum = 0
FOR gi = 0 TO 8000
sum = sum + W(gi) / many
IF sum > u(uu) THEN GOTO 45
NEXT gi
45 PRINT USING "##.### .### "; gi / 1000; sum
NEXT uu
END



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