Math Forum Discussions

Luis A. Afonso

Posts: 4,758
From: LIsbon (Portugal)
Registered: 2/16/05

Re: Kolmogorov-Smirnov-Lilliefors Test statistics
Posted: Sep 20, 2012 1:49 PM

Plain Text

In testing for normality using the Lilliefors (Kolmagorov-Smirnov) tough we get an anomalous high test statistics, D max, we stay pretty sure that data the null Hypotheses should be rejected.
This obvious decision will be illustrated numerically trough a few examples where simulated Fish samples,
______cdf = 1/( 1- x^(-2))
were tested. Recording the max among the tested samples. We had found:

Size n=100/400000____Fisk______Normal_
__________________ 728.36_____0.239__
_________/40000___ 504.98_____0.221
__________________ 412.05_____0.226
__________________ 487.50_____0.230

It can be added that in [.2, .3) the frequencies for Normal data are very low, sometimes 0.0000, or 0.0005 say: in fact almost all test statistics are contained in [0, .2).
No surprise . . . at all.

Luis A. Afonso

REM "FIG"
CLS
PRINT " ************** FIG "
PRINT " ********************"
DEFDBL A-Z
PRINT " "
PRINT " "
REM
INPUT " SAMPLE SIZE "; N
INPUT " HOW MANY "; ali
INPUT " 0= N or 1= Fisk "; wnm
DIM x(N), xx(N), f(N), diff(N), h(20)
DEF fng (z, j) = -.5 * z ^ 2 * (2 * j + 1) / ((j + 1) * (2 * j + 2))
f(0) = 0
suptest = 0: infra = 1E+20
FOR ji = 1 TO N: f(ji) = ji / N: NEXT ji
REM THE CHI2 TEST:
REM
REM
pi = 4 * ATN(1): c = 1 / SQR(2 * pi)
REM
FOR SAMPLE = 1 TO ali
RANDOMIZE TIMER
mmajor = 0
LOCATE 8, 50: PRINT USING "########"; ali - SAMPLE
md = 0: sum2 = 0
FOR item = 1 TO N
14 h = RND
IF h < 1E-14 THEN GOTO 14
IF wnm = 0 THEN GOTO 1
COLOR 14: u = 1 / SQR(1 / h - 1)
GOTO 2
1 COLOR 3: aa = SQR(-2 * LOG(h))
u = 0 + 1 * aa * COS(2 * pi * RND)
2 x(item) = u
md = md + u / N
sum2 = sum2 + u * u
NEXT item
sqd = sum2 - N * (md ^ 2)
sd = SQR(sqd / (N - 1))
FOR ii = 1 TO N: x(ii) = (x(ii) - md) / sd: NEXT ii
REM Ordering
FOR i1 = 1 TO N: u = x(i1): w = 1
FOR i2 = 1 TO N
IF x(i2) < u THEN w = w + 1
NEXT i2: xx(w) = u
NEXT i1
REM "**********************"
REM PHI
FOR tt = 1 TO N: z = xx(tt)
IF z >= 0 THEN kw = 0
IF z < 0 THEN kw = 1
zu = ABS(z): s = c * zu: antes = c * zu
FOR j = 1 TO 10000
xx = antes * fng(zu, j)
s = s + xx
antes = xx
IF ABS(xx) < .00005 THEN GOTO 20
NEXT j
20 IF kw = 0 THEN FF = .5 + s
IF kw = 1 THEN FF = .5 - s
b = ABS(FF - f(tt - 1))
bb = ABS(f(tt) - FF)
REM
MAJOR = b
IF bb > b THEN MAJOR = bb
REM
IF MAJOR > mmajor THEN mmajor = MAJOR
IF mmajor > suptest THEN suptest = mmajor
IF mmajor < infra THEN infra = mmajor
NEXT tt
FOR tu = 1 TO N
IF mmajor > 0 AND mmajor < .1 THEN h(1) = h(1) + 1
IF mmajor >= .1 AND mmajor < .2 THEN h(2) = h(2) + 1
IF mmajor >= .2 AND mmajor < .3 THEN h(3) = h(3) + 1
IF mmajor >= .3 AND mmajor < .4 THEN h(4) = h(4) + 1
IF mmajor >= .4 AND mmajor < .5 THEN h(5) = h(5) + 1
IF mmajor >= .5 AND mmajor < .6 THEN h(6) = h(6) + 1
IF mmajor >= .6 AND mmajor < .7 THEN h(7) = h(7) + 1
IF mmajor >= .7 AND mmajor < .8 THEN h(8) = h(8) + 1
IF mmajor >= .8 AND mmajor < .9 THEN h(9) = h(9) + 1
IF mmajor >= .9 AND mmajor < 1 THEN h(10) = h(10) + 1
NEXT tu
NEXT SAMPLE
FOR ty = 1 TO 10
LOCATE 10 + ty, 7
PRINT USING "#.####"; h(ty) / (ali * N)
NEXT ty
PRINT " suptest "; suptest, infra
100 END