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/* Re Kolmogorov statistics, here is Birnbaum and Tingey's formula for the
distribution of D+, the maximum of all positive deviations between a
theoretical distribution function P(x) and an empirical one Sn(x)
from n samples.
+
D = sup [P(x) - S (x)]
n -inf < x < inf n
[n(1-e)]
+ - v-1 n-v
Pr{D > e} = > C e (e + v/n) (1 - e - v/n)
n - n v
v=0
[n(1-e)] is the largest integer not exceeding n(1-e).
nCv is the number of combinations of n things taken v at a time. */
#include <math.h>
#ifdef ANSIPROT
extern double pow ( double, double );
extern double floor ( double );
extern double lgam ( double );
extern double exp ( double );
extern double sqrt ( double );
extern double log ( double );
extern double fabs ( double );
double smirnov ( int, double );
double kolmogorov ( double );
#else
double pow (), floor (), lgam (), exp (), sqrt (), log (), fabs ();
double smirnov (), kolmogorov ();
#endif
extern double MAXLOG;
/* Exact Smirnov statistic, for one-sided test. */
double
smirnov (n, e)
int n;
double e;
{
int v, nn;
double evn, omevn, p, t, c, lgamnp1;
if (n <= 0 || e < 0.0 || e > 1.0)
return (-1.0);
nn = floor ((double) n * (1.0 - e));
p = 0.0;
if (n < 1013)
{
c = 1.0;
for (v = 0; v <= nn; v++)
{
evn = e + ((double) v) / n;
p += c * pow (evn, (double) (v - 1))
* pow (1.0 - evn, (double) (n - v));
/* Next combinatorial term; worst case error = 4e-15. */
c *= ((double) (n - v)) / (v + 1);
}
}
else
{
lgamnp1 = lgam ((double) (n + 1));
for (v = 0; v <= nn; v++)
{
evn = e + ((double) v) / n;
omevn = 1.0 - evn;
if (fabs (omevn) > 0.0)
{
t = lgamnp1
- lgam ((double) (v + 1))
- lgam ((double) (n - v + 1))
+ (v - 1) * log (evn)
+ (n - v) * log (omevn);
if (t > -MAXLOG)
p += exp (t);
}
}
}
return (p * e);
}
/* Kolmogorov's limiting distribution of two-sided test, returns
probability that sqrt(n) * max deviation > y,
or that max deviation > y/sqrt(n).
The approximation is useful for the tail of the distribution
when n is large. */
double
kolmogorov (y)
double y;
{
double p, t, r, sign, x;
x = -2.0 * y * y;
sign = 1.0;
p = 0.0;
r = 1.0;
do
{
t = exp (x * r * r);
p += sign * t;
if (t == 0.0)
break;
r += 1.0;
sign = -sign;
}
while ((t / p) > 1.1e-16);
return (p + p);
}
/* Functional inverse of Smirnov distribution
finds e such that smirnov(n,e) = p. */
double
smirnovi (n, p)
int n;
double p;
{
double e, t, dpde;
if (p <= 0.0 || p > 1.0)
{
mtherr ("smirnovi", DOMAIN);
return 0.0;
}
/* Start with approximation p = exp(-2 n e^2). */
e = sqrt (-log (p) / (2.0 * n));
do
{
/* Use approximate derivative in Newton iteration. */
t = -2.0 * n * e;
dpde = 2.0 * t * exp (t * e);
if (fabs (dpde) > 0.0)
t = (p - smirnov (n, e)) / dpde;
else
{
mtherr ("smirnovi", UNDERFLOW);
return 0.0;
}
e = e + t;
if (e >= 1.0 || e <= 0.0)
{
mtherr ("smirnovi", OVERFLOW);
return 0.0;
}
}
while (fabs (t / e) > 1e-10);
return (e);
}
/* Functional inverse of Kolmogorov statistic for two-sided test.
Finds y such that kolmogorov(y) = p.
If e = smirnovi (n,p), then kolmogi(2 * p) / sqrt(n) should
be close to e. */
double
kolmogi (p)
double p;
{
double y, t, dpdy;
if (p <= 0.0 || p > 1.0)
{
mtherr ("kolmogi", DOMAIN);
return 0.0;
}
/* Start with approximation p = 2 exp(-2 y^2). */
y = sqrt (-0.5 * log (0.5 * p));
do
{
/* Use approximate derivative in Newton iteration. */
t = -2.0 * y;
dpdy = 4.0 * t * exp (t * y);
if (fabs (dpdy) > 0.0)
t = (p - kolmogorov (y)) / dpdy;
else
{
mtherr ("kolmogi", UNDERFLOW);
return 0.0;
}
y = y + t;
}
while (fabs (t / y) > 1e-10);
return (y);
}
#ifdef SALONE
/* Type in a number. */
void
getnum (s, px)
char *s;
double *px;
{
char str[30];
printf (" %s (%.15e) ? ", s, *px);
gets (str);
if (str[0] == '\0' || str[0] == '\n')
return;
sscanf (str, "%lf", px);
printf ("%.15e\n", *px);
}
/* Type in values, get answers. */
void
main ()
{
int n;
double e, p, ps, pk, ek, y;
n = 5;
e = 0.0;
p = 0.1;
loop:
ps = n;
getnum ("n", &ps);
n = ps;
if (n <= 0)
{
printf ("? Operator error.\n");
goto loop;
}
/*
getnum ("e", &e);
ps = smirnov (n, e);
y = sqrt ((double) n) * e;
printf ("y = %.4e\n", y);
pk = kolmogorov (y);
printf ("Smirnov = %.15e, Kolmogorov/2 = %.15e\n", ps, pk / 2.0);
*/
getnum ("p", &p);
e = smirnovi (n, p);
printf ("Smirnov e = %.15e\n", e);
y = kolmogi (2.0 * p);
ek = y / sqrt ((double) n);
printf ("Kolmogorov e = %.15e\n", ek);
goto loop;
}
#endif
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