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

A not expected asymptotical test statisticsI 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 meansen.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))WithssdX = 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__________ 97.7The 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 = 0REM        FOR t = 1 TO n1       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 + 140      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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