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Topic: Kolmogorov-Smirnov-Lilliefors Test statistics
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Luis A. Afonso

Posts: 4,617
From: LIsbon (Portugal)
Registered: 2/16/05
Kolmogorov-Smirnov-Lilliefors Test statistics
Posted: Sep 16, 2012 5:57 PM
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Kolmogorov-Smirnov-Lilliefors Test statistics


The maxima and secondary maxima discrepancy from Normal data: Critical Values.

______________________________________________

Lilliefors Test Table (1million samples/size)
The First and Second maximum differences between the Normal Distribution values and sample size n cumulative fractiles. M.A. Molin & Abdil values.

citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.64.5753&am

Fractiles:
______0.80____0.85____0.90___ 0.95____0.99__

__n=5
_____0.2163__0.2246__0.2351__0.2498__0.2767__2nd
_____0.2895__0.3028__0.3190__0.3431__0.3964__1st
_____0.2893__0.3027__0.3188__0.3427__0.3959__M.A.

__n=10
_____0.1761__0.1843__0.1950__0.2123__0.2475__
_____0.2171__0.2273__0.2410__0.2622__0.3035__
_____0.2171__0.2273__0.2410__0.2616__0.3037__

__n=15
_____0.1534__0.1609__0.1709__0.1866__0.2191__
_____0.1845__0.1899__0.2014__0.2192__0.2542__
_____0.1811__0.1899__0.2016__0.2196__0.2245__

__n=20
_____0.1379__0.1448__0.1538__0.1682__0.1975__
_____0.1588__0.1664__0.1763__0.1918__0.2236__
_____0.1589__0.1666__0.1764__0.1920__0.2226__

__n=25
_____0.1263__0.1327__0.1411__0.1543__0.1815__
_____0.1430__0.1498__0.1589__0.1731__0.2015__
_____0.1429__0.1498__0.1589__0.1726__0.2010__

__n=30
_____0.1174__0.1233__0.1311__0.1434__0.1686__
_____0.1313__0.1376__0.1458__0.1588__0.1849__
_____0.1315__0.1378__0.1460__0.1590__0.1848__

__n=35
_____0.1101__0.1157__0.1230__0.1346__0.1582__
_____0.1220__0.1279__0.1356__0.1475__0.1719__
_____0.1220__0.1278__0.1356__0.1478__0.1720__

__n=40
_____0.1041__0.1094__0.1163__0.1272__0.1495__
_____0.1144__0.1200__0.1272__0.1384__0.1614__
_____0.1147__0.1204__0.1275__0.1386__0.1616__

__n=45
_____0.0991__0.1041__0.1107__0.1211__0.1422__
_____0.1082__0.1135__0.1203__0.1310__0.1527__
_____0.1083__0.1134__0.1204__0.1309__0.1525__

__n=50
_____0.0946__0.0994__0.1057__0.1155__0.1355__
_____0.1028__0.1078__0.1143__0.1244__0.1448__
_____0.1030__0.1079__0.1142__0.1246__0.1457__

Explanation

The third lines (M.A.) show the Morin/Abdil values, the remaining two, mine: the first records the sub maxima, the second the KS-Lilliefors maxima. It can be seen easily that there is an excellent agreement between lines which demonstrate that my simulations got correct values. It should be stresses that I simulate 1 million samples by sample size and them 100 000. It?s true that the fourth decimal places are absolutely abusive in view the forecasted random errors.
We intend to explore a little how informative (if so) is this *second* maximum in view the conclusions to draw from tests.
The program ?New? listing follows.


Luis A. Afonso


REM "NEW"
CLS
PRINT : PRINT
COLOR 12
PRINT " 1st maximum and 2nd KOLMOGOROV - SMIRNOV -";
PRINT "LILLIEFORS test statistics "
INPUT " n (SAMPLE SIZE) = "; n
INPUT " HOW MANY SAMPLES = "; ali
pi = 4 * ATN(1): c = 1 / SQR(2 * pi)
DIM x(n), xx(n), F(n), y(n)
DIM max(9001), max2(9001)
DEF fng (z, j) = -.5 * z ^ 2 * (2 * j + 1) / ((j + 1) * (2 * j + 3))
F(0) = 0
FOR ji = 0 TO n: F(ji) = ji / n: NEXT ji
REM
REM
REM k = SAMPLE
FOR k = 1 TO ali: RANDOMIZE TIMER
mmajor = -1E-20: second = mmajor
LOCATE 7, 50: PRINT USING "##########"; ali - k
md = 0: sum2 = 0
REM
FOR i = 1 TO n
a = SQR(-2 * LOG(RND))
x(i) = a * COS(2 * pi * RND)
md = md + x(i) / n
sum2 = sum2 + x(i) * x(i)
NEXT i
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 ii = 1 TO n: u = x(ii): W = 1
FOR jj = 1 TO n
IF x(jj) < u THEN W = W + 1
NEXT jj: xx(W) = u
NEXT ii
REM "******************"
REM PHI CALCULATION
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 = 0 TO 100000
xx = antes * fng(zu, j)
s = s + xx
antes = xx
IF ABS(xx) < .00004 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)
maior = b
IF bb > b THEN maior = bb
x(tt) = maior
GOTO 99
REM local difference= x(tt)
99 REM
NEXT tt
higher = -1
FOR ii = 1 TO n
IF x(ii) <= higher THEN GOTO 22
higher = x(ii): llocal = ii
22 NEXT ii
LOCATE 10, 1
x(llocal) = -2: high = -1
FOR i2 = 1 TO n
IF x(i2) <= high THEN GOTO 33
high = x(i2): ll = i2
33 NEXT i2
higher = INT(10000 * higher + .5)
IF higher > 8000 THEN higher = 8000
high = INT(10000 * high + .5)
IF high > 9000 THEN high = 9000
max(higher) = max(higher) + 1 / ali
max2(high) = max2(high) + 1 / ali
NEXT k
COLOR 14
LOCATE 8, 10
PRINT " largest 2nd "
c(0) = .8: c(1) = .85: c(2) = .9
c(3) = .95: c(4) = .99
FOR tu = 0 TO 4: smax = 0
FOR zx = 0 TO 8000
smax = smax + max(zx)
IF smax > c(tu) THEN GOTO 4
NEXT zx
4 LOCATE 10 + tu, 15
PRINT USING "#.#### "; zx / 10000; smax
NEXT tu: smax = 0
FOR tu = 0 TO 4: smax = 0
FOR zx = 0 TO 8000
smax = smax + max2(zx)
IF smax > c(tu) THEN GOTO 5
NEXT zx
5 LOCATE 10 + tu, 45
PRINT USING "#.#### "; zx / 10000; smax
NEXT tu
COLOR 7
END



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