1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
|
/* cdf/inverse_normal.c
*
* Copyright (C) 2002 Przemyslaw Sliwa and Jason H. Stover.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or (at
* your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA.
*/
/*
* Computes the inverse normal cumulative distribution function
* according to the algorithm shown in
*
* Wichura, M.J. (1988).
* Algorithm AS 241: The Percentage Points of the Normal Distribution.
* Applied Statistics, 37, 477-484.
*/
#include <config.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_cdf.h>
#include "rat_eval.h"
static double
small (double q)
{
const double a[8] = { 3.387132872796366608, 133.14166789178437745,
1971.5909503065514427, 13731.693765509461125,
45921.953931549871457, 67265.770927008700853,
33430.575583588128105, 2509.0809287301226727
};
const double b[8] = { 1.0, 42.313330701600911252,
687.1870074920579083, 5394.1960214247511077,
21213.794301586595867, 39307.89580009271061,
28729.085735721942674, 5226.495278852854561
};
double r = 0.180625 - q * q;
double x = q * rat_eval (a, 8, b, 8, r);
return x;
}
static double
intermediate (double r)
{
const double a[] = { 1.42343711074968357734, 4.6303378461565452959,
5.7694972214606914055, 3.64784832476320460504,
1.27045825245236838258, 0.24178072517745061177,
0.0227238449892691845833, 7.7454501427834140764e-4
};
const double b[] = { 1.0, 2.05319162663775882187,
1.6763848301838038494, 0.68976733498510000455,
0.14810397642748007459, 0.0151986665636164571966,
5.475938084995344946e-4, 1.05075007164441684324e-9
};
double x = rat_eval (a, 8, b, 8, (r - 1.6));
return x;
}
static double
tail (double r)
{
const double a[] = { 6.6579046435011037772, 5.4637849111641143699,
1.7848265399172913358, 0.29656057182850489123,
0.026532189526576123093, 0.0012426609473880784386,
2.71155556874348757815e-5, 2.01033439929228813265e-7
};
const double b[] = { 1.0, 0.59983220655588793769,
0.13692988092273580531, 0.0148753612908506148525,
7.868691311456132591e-4, 1.8463183175100546818e-5,
1.4215117583164458887e-7, 2.04426310338993978564e-15
};
double x = rat_eval (a, 8, b, 8, (r - 5.0));
return x;
}
double
gsl_cdf_ugaussian_Pinv (const double P)
{
double r, x, pp;
double dP = P - 0.5;
if (P == 1.0)
{
return GSL_POSINF;
}
else if (P == 0.0)
{
return GSL_NEGINF;
}
if (fabs (dP) <= 0.425)
{
x = small (dP);
return x;
}
pp = (P < 0.5) ? P : 1.0 - P;
r = sqrt (-log (pp));
if (r <= 5.0)
{
x = intermediate (r);
}
else
{
x = tail (r);
}
if (P < 0.5)
{
return -x;
}
else
{
return x;
}
}
double
gsl_cdf_ugaussian_Qinv (const double Q)
{
double r, x, pp;
double dQ = Q - 0.5;
if (Q == 1.0)
{
return GSL_NEGINF;
}
else if (Q == 0.0)
{
return GSL_POSINF;
}
if (fabs (dQ) <= 0.425)
{
x = small (dQ);
return -x;
}
pp = (Q < 0.5) ? Q : 1.0 - Q;
r = sqrt (-log (pp));
if (r <= 5.0)
{
x = intermediate (r);
}
else
{
x = tail (r);
}
if (Q < 0.5)
{
return x;
}
else
{
return -x;
}
}
double
gsl_cdf_gaussian_Pinv (const double P, const double sigma)
{
return sigma * gsl_cdf_ugaussian_Pinv (P);
}
double
gsl_cdf_gaussian_Qinv (const double Q, const double sigma)
{
return sigma * gsl_cdf_ugaussian_Qinv (Q);
}
|
