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Luis A. Afonso
Posts:
4,277
From:
LIsbon (Portugal)
Registered:
2/16/05
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A not expected asymptotical test statistics
Posted:
Nov 20, 2012 12:08 PM
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A not expected asymptotical test statistics
I feel I was rewarded, this time, by my curiosity to search for ill-conformations. Trying to illustrate a current test by numeric examples, I was able to show that an asymptotical characteristic was present, the text-books omit, as far as I know.
For D we note the difference of homoscedastic Population normal means
en.wikipedia.org/wiki/Homoscedasticity
we are dealing with the Populations noted by X~N(muX, sigma):nX, Y~N(muX + D, sigma):nY. For example:
_______ X ~ N(D, 1):20, Y ~ N(0, 1):20, D>=0.
The test statistics T is such that the following interval contains D with 95% probability
(xhat-yhat) - 2.02439* nw <= D <=
<= (xhat-yhat) + 2.02439* nw
__ nw= sqrt ( (ssdX+ssdY) * (1/nX+1/nY)/(nX+nY-2))
With
ssdX = sum squares differences; sample X, size 20, which mean is xhat. (Similarly for sample Y); nX=nY=20.
__________________________
_Results (program PW)
_______n=6 ____2.228___(10df)
______ D= 0____Power= 100 %
_________ 2__________ 98.8
_________ 4__________ 97.3
__________6__________ 96.7
_______n=14 ____2.056___(26df)
______ D= 0____Power= 97.4%
_________ 2__________ 97.2
_________ 4__________ 97.7
_________10__________ 98.4
_______n=20____2.02439___(38df)
______ D= 0____Power= 96.7 %
_________ 2__________ 96.7
_________ 4__________ 97.0
__________6__________ 97.5
_______n=50____1.98447___(98df)
______ D= 0____Power= 95.6%
_________ 2__________ 95.9
_________ 4__________ 96.8
_________ 6__________ 97.7
_______n=100____1.97202__(198df)
______ D= 0____Power= 95.4%
__________2__________ 95.6
__________4__________ 96.5
__________6__________ 97.7
The expectation is that D=0 provide a 95% Power, which grows to 100% with increasing D, the Population X mean value differs from 0. It?s really what I?d found. However it?s odd that the interval between bounds for very small sizes, 6x6, contains, wrongly, as much as 100% sample T´s, when exactly 95% was expected by construction: two tails test, 5% confidence level. The only explanation is that the tests statistics frequencies that are outside the *acceptance* interval only asymptotically (i.e. for large sample sizes) tends to Student distribution probability 1-alpha/2.
Luis A. Afonso
REM "PW"
CLS
PRINT
PRINT "______________PW__________________"
DEFDBL A-Z
PRINT " POWER (95% CI) for D>=0 ";
PRINT " X~N(D,1):n, Y~N(0,1):n ";
PRINT " 6+ 6=T0=2.228___14+14=2.056__20+20=2.02439"
PRINT " 30+30=2.00172__40+40=1.99085__50+50=1.98447"
PRINT " 60+60=1.98027__70+70=1.97730__80+80=1.97509"
PRINT " 90+90=1.97338__100+100=1.97202 "
INPUT " nX=nY , T0 "; n, T0
INPUT " how many "; many
nw = (1 / n + 1 / n) / (n + n - 2)
pi = 4 * ATN(1)
FOR ti = 0 TO 20 STEP 2
D = ti
RANDOMIZE TIMER
FOR i = 1 TO many
LOCATE 10, 30
PRINT USING "#########"; many - i
PRINT " D POWER% T --> "
sx = 0: sy = 0: ssx = 0: ssy = 0
REM
FOR t = 1 TO n
1 a = RND
IF a < 1E-15 THEN GOTO 1
a = SQR(-2 * LOG(a))
x = D + 10 * a * COS(2 * pi * RND)
sx = sx + x
ssx = ssx + x * x
y = 0 + a * SIN(2 * pi * RND)
sy = sy + y
ssy = ssy + y * y
NEXT t
mx = sx / n
sdx = ssx - n * mx * mx
my = sy / n
sdy = ssy - n * my * my
u = T0 * SQR((ssx + ssy) * nw)
u0 = (mx - my) - u: u1 = (mx - my) + u
COLOR 7
IF u0 < D AND u1 > D THEN GOTO 40
rj = rj + 1
40 NEXT i
w = w + 1
LOCATE 10 + w, 30
PRINT USING "##"; D;
PRINT USING " ###.# #.### "; 100 * (1 - rj / i); T0
rj = 0
NEXT ti
END
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