Date: Sep 16, 2012 5:57 PM Author: Luis A. Afonso Subject: Kolmogorov-Smirnov-Lilliefors Test statistics Kolmogorov-Smirnov-Lilliefors Test statistics

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

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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