C KLMO 10
C ..................................................................KLMO 20
C KLMO 30
C SUBROUTINE KOLMO KLMO 40
C KLMO 50
C PURPOSE KLMO 60
C TESTS THE DIFFERENCE BETWEEN EMPIRICAL AND THEORETICAL KLMO 70
C DISTRIBUTIONS USING THE KOLMOGOROV-SMIRNOV TEST KLMO 80
C KLMO 90
C USAGE KLMO 100
C CALL KOLMO(X,N,Z,PROB,IFCOD,U,S,IER) KLMO 110
C KLMO 120
C DESCRIPTION OF PARAMETERS KLMO 130
C X - INPUT VECTOR OF N INDEPENDENT OBSERVATIONS. ON KLMO 140
C RETURN FROM KOLMO, X HAS BEEN SORTED INTO A KLMO 150
C MONOTONIC NON-DECREASING SEQUENCE. KLMO 160
C N - NUMBER OF OBSERVATIONS IN X KLMO 170
C Z - OUTPUT VARIABLE CONTAINING THE GREATEST VALUE WITH KLMO 180
C RESPECT TO X OF SQRT(N)*ABS(FN(X)-F(X)) WHERE KLMO 190
C F(X) IS A THEORETICAL DISTRIBUTION FUNCTION AND KLMO 200
C FN(X) AN EMPIRICAL DISTRIBUTION FUNCTION. KLMO 210
C PROB - OUTPUT VARIABLE CONTAINING THE PROBABILITY OF KLMO 220
C THE STATISTIC BEING GREATER THAN OR EQUAL TO Z IF KLMO 230
C THE HYPOTHESIS THAT X IS FROM THE DENSITY UNDER KLMO 240
C CONSIDERATION IS TRUE. E.G., PROB = 0.05 IMPLIES KLMO 250
C THAT ONE CAN REJECT THE NULL HYPOTHESIS THAT THE SET KLMO 260
C X IS FROM THE DENSITY UNDER CONSIDERATION WITH 5 PER KLMO 270
C CENT PROBABILITY OF BEING INCORRECT. PROB = 1. - KLMO 280
C SMIRN(Z). KLMO 290
C IFCOD- A CODE DENOTING THE PARTICULAR THEORETICAL KLMO 300
C PROBABILITY DISTRIBUTION FUNCTION BEING CONSIDERED. KLMO 310
C = 1---F(X) IS THE NORMAL PDF. KLMO 320
C = 2---F(X) IS THE EXPONENTIAL PDF. KLMO 330
C = 3---F(X) IS THE CAUCHY PDF. KLMO 340
C = 4---F(X) IS THE UNIFORM PDF. KLMO 350
C = 5---F(X) IS USER SUPPLIED. KLMO 360
C U - WHEN IFCOD IS 1 OR 2, U IS THE MEAN OF THE DENSITY KLMO 370
C GIVEN ABOVE. KLMO 380
C WHEN IFCOD IS 3, U IS THE MEDIAN OF THE CAUCHY KLMO 390
C DENSITY. KLMO 400
C WHEN IFCOD IS 4, U IS THE LEFT ENDPOINT OF THE KLMO 410
C UNIFORM DENSITY. KLMO 420
C WHEN IFCOD IS 5, U IS USER SPECIFIED. KLMO 430
C S - WHEN IFCOD IS 1 OR 2, S IS THE STANDARD DEVIATION OF KLMO 440
C DENSITY GIVEN ABOVE, AND SHOULD BE POSITIVE. KLMO 450
C WHEN IFCOD IS 3, U - S SPECIFIES THE FIRST QUARTILE KLMO 460
C OF THE CAUCHY DENSITY. S SHOULD BE NON-ZERO. KLMO 470
C IF IFCOD IS 4, S IS THE RIGHT ENDPOINT OF THE UNIFORMKLMO 480
C DENSITY. S SHOULD BE GREATER THAN U. KLMO 490
C IF IFCOD IS 5, S IS USER SPECIFIED. KLMO 500
C IER - ERROR INDICATOR WHICH IS NON-ZERO IF S VIOLATES ABOVEKLMO 510
C CONVENTIONS. ON RETURN NO TEST HAS BEEN MADE, AND X KLMO 520
C AND Y HAVE BEEN SORTED INTO MONOTONIC NON-DECREASING KLMO 530
C SEQUENCES. IER IS SET TO ZERO ON ENTRY TO KOLMO. KLMO 540
C IER IS CURRENTLY SET TO ONE IF THE USER-SUPPLIED PDF KLMO 550
C IS REQUESTED FOR TESTING. THIS SHOULD BE CHANGED KLMO 560
C (SEE REMARKS) WHEN SOME PDF IS SUPPLIED BY THE USER. KLMO 570
C KLMO 580
C REMARKS KLMO 590
C N SHOULD BE GREATER THAN OR EQUAL TO 100. (SEE THE KLMO 600
C MATHEMATICAL DESCRIPTION GIVEN FOR THE PROGRAM SMIRN, KLMO 610
C CONCERNING ASYMPTOTIC FORMULAE) ALSO, PROBABILITY LEVELS KLMO 620
C DETERMINED BY THIS PROGRAM WILL NOT BE CORRECT IF THE KLMO 630
C SAME SAMPLES ARE USED TO ESTIMATE PARAMETERS FOR THE KLMO 640
C CONTINUOUS DISTRIBUTIONS WHICH ARE USED IN THIS TEST. KLMO 650
C (SEE THE MATHEMATICAL DESCRIPTION FOR THIS PROGRAM) KLMO 660
C F(X) SHOULD BE A CONTINUOUS FUNCTION. KLMO 670
C ANY USER SUPPLIED CUMULATIVE PROBABILITY DISTRIBUTION KLMO 680
C FUNCTION SHOULD BE CODED BEGINNING WITH STATEMENT 26 BELOW, KLMO 690
C AND SHOULD RETURN TO STATEMENT 27. KLMO 700
C KLMO 710
C DOUBLE PRECISION USAGE---IT IS DOUBTFUL THAT THE USER WILL KLMO 720
C WISH TO PERFORM THIS TEST USING DOUBLE PRECISION ACCURACY. KLMO 730
C IF ONE WISHES TO COMMUNICATE WITH KOLMO IN A DOUBLE KLMO 740
C PRECISION PROGRAM, HE SHOULD CALL THE FORTRAN SUPPLIED KLMO 750
C PROGRAM SNGL(X) PRIOR TO CALLING KOLMO, AND CALL THE KLMO 760
C FORTRAN SUPPLIED PROGRAM DBLE(X) AFTER EXITING FROM KOLMO. KLMO 770
C (NOTE THAT SUBROUTINE SMIRN DOES HAVE DOUBLE PRECISION KLMO 780
C CAPABILITY AS SUPPLIED BY THIS PACKAGE.) KLMO 790
C KLMO 800
C KLMO 810
C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED KLMO 820
C SMIRN, NDTR, AND ANY USER SUPPLIED SUBROUTINES REQUIRED. KLMO 830
C KLMO 840
C METHOD KLMO 850
C FOR REFERENCE, SEE (1) W. FELLER--ON THE KOLMOGOROV-SMIRNOV KLMO 860
C LIMIT THEOREMS FOR EMPIRICAL DISTRIBUTIONS-- KLMO 870
C ANNALS OF MATH. STAT., 19, 1948. 177-189, KLMO 880
C (2) N. SMIRNOV--TABLE FOR ESTIMATING THE GOODNESS OF FIT KLMO 890
C OF EMPIRICAL DISTRIBUTIONS--ANNALS OF MATH. STAT., 19, KLMO 900
C 1948. 279-281. KLMO 910
C (3) R. VON MISES--MATHEMATICAL THEORY OF PROBABILITY AND KLMO 920
C STATISTICS--ACADEMIC PRESS, NEW YORK, 1964. 490-493, KLMO 930
C (4) B.V. GNEDENKO--THE THEORY OF PROBABILITY--CHELSEA KLMO 940
C PUBLISHING COMPANY, NEW YORK, 1962. 384-401. KLMO 950
C KLMO 960
C ..................................................................KLMO 970
C KLMO 980
SUBROUTINE KOLMO(X,N,Z,PROB,IFCOD,U,S,IER) KLMO 990
DIMENSION X(1) KLMO1000
C KLMO1010
C NON DECREASING ORDERING OF X(I)'S (DUBY METHOD) KLMO1020
C KLMO1030
IER=0 KLMO1040
DO 5 I=2,N KLMO1050
IF(X(I)-X(I-1))1,5,5 KLMO1060
1 TEMP=X(I) KLMO1070
IM=I-1 KLMO1080
DO 3 J=1,IM KLMO1090
L=I-J KLMO1100
IF(TEMP-X(L))2,4,4 KLMO1110
2 X(L+1)=X(L) KLMO1120
3 CONTINUE KLMO1130
X(1)=TEMP KLMO1140
GO TO 5 KLMO1150
4 X(L+1)=TEMP KLMO1160
5 CONTINUE KLMO1170
C KLMO1180
C COMPUTES MAXIMUM DEVIATION DN IN ABSOLUTE VALUE BETWEEN KLMO1190
C EMPIRICAL AND THEORETICAL DISTRIBUTIONS KLMO1200
C KLMO1210
NM1=N-1 KLMO1220
XN=N KLMO1230
DN=0.0 KLMO1240
FS=0.0 KLMO1250
IL=1 KLMO1260
6 DO 7 I=IL,NM1 KLMO1270
J=I KLMO1280
IF(X(J)-X(J+1))9,7,9 KLMO1290
7 CONTINUE KLMO1300
8 J=N KLMO1310
9 IL=J+1 KLMO1320
FI=FS KLMO1330
FS=FLOAT(J)/XN KLMO1340
IF(IFCOD-2)10,13,17 KLMO1350
10 IF(S)11,11,12 KLMO1360
11 IER=1 KLMO1370
GO TO 29 KLMO1380
12 Z =(X(J)-U)/S KLMO1390
CALL NDTR(Z,Y,D) KLMO1400
GO TO 27 KLMO1410
13 IF(S)11,11,14 KLMO1420
14 Z=(X(J)-U)/S+1.0 KLMO1430
IF(Z)15,15,16 KLMO1440
15 Y=0.0 KLMO1450
GO TO 27 KLMO1460
16 Y=1.-EXP(-Z) KLMO1470
GO TO 27 KLMO1480
17 IF(IFCOD-4)18,20,26 KLMO1490
18 IF(S)19,11,19 KLMO1500
19 Y=ATAN((X(J)-U)/S)*0.3183099+0.5 KLMO1510
GO TO 27 KLMO1520
20 IF(S-U)11,11,21 KLMO1530
21 IF(X(J)-U)22,22,23 KLMO1540
22 Y=0.0 KLMO1550
GO TO 27 KLMO1560
23 IF(X(J)-S)25,25,24 KLMO1570
24 Y=1.0 KLMO1580
GO TO 27 KLMO1590
25 Y=(X(J)-U)/(S-U) KLMO1600
GO TO 27 KLMO1610
26 IER=1 KLMO1620
GO TO 29 KLMO1630
27 EI=ABS(Y-FI) KLMO1640
ES=ABS(Y-FS) KLMO1650
DN=AMAX1(DN,EI,ES) KLMO1660
IF(IL-N)6,8,28 KLMO1670
C KLMO1680
C COMPUTES Z=DN*SQRT(N) AND PROBABILITY KLMO1690
C KLMO1700
28 Z=DN*SQRT(XN) KLMO1710
CALL SMIRN(Z,PROB) KLMO1720
PROB=1.0-PROB KLMO1730
29 RETURN KLMO1740
END KLMO1750