Date: Nov 20, 2012 12:08 PM Author: Luis A. Afonso Subject: A not expected asymptotical test statistics 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 "______________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