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
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
|
/* stdtr.c
*
* Student's t distribution
*
*
*
* SYNOPSIS:
*
* double t, stdtr();
* short k;
*
* y = stdtr( k, t );
*
*
* DESCRIPTION:
*
* Computes the integral from minus infinity to t of the Student
* t distribution with integer k > 0 degrees of freedom:
*
* t
* -
* | |
* - | 2 -(k+1)/2
* | ( (k+1)/2 ) | ( x )
* ---------------------- | ( 1 + --- ) dx
* - | ( k )
* sqrt( k pi ) | ( k/2 ) |
* | |
* -
* -inf.
*
* Relation to incomplete beta integral:
*
* 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
* where
* z = k/(k + t**2).
*
* For t < -2, this is the method of computation. For higher t,
* a direct method is derived from integration by parts.
* Since the function is symmetric about t=0, the area under the
* right tail of the density is found by calling the function
* with -t instead of t.
*
* ACCURACY:
*
* Tested at random 1 <= k <= 25. The "domain" refers to t.
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -100,-2 50000 5.9e-15 1.4e-15
* IEEE -2,100 500000 2.7e-15 4.9e-17
*/
/* stdtri.c
*
* Functional inverse of Student's t distribution
*
*
*
* SYNOPSIS:
*
* double p, t, stdtri();
* int k;
*
* t = stdtri( k, p );
*
*
* DESCRIPTION:
*
* Given probability p, finds the argument t such that stdtr(k,t)
* is equal to p.
*
* ACCURACY:
*
* Tested at random 1 <= k <= 100. The "domain" refers to p:
* Relative error:
* arithmetic domain # trials peak rms
* IEEE .001,.999 25000 5.7e-15 8.0e-16
* IEEE 10^-6,.001 25000 2.0e-12 2.9e-14
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*/
#include <math.h>
extern double PI, MACHEP, MAXNUM;
#ifdef ANSIPROT
extern double sqrt ( double );
extern double atan ( double );
extern double incbet ( double, double, double );
extern double incbi ( double, double, double );
extern double fabs ( double );
#else
double sqrt(), atan(), incbet(), incbi(), fabs();
#endif
double stdtr( k, t )
int k;
double t;
{
double x, rk, z, f, tz, p, xsqk;
int j;
if( k <= 0 )
{
mtherr( "stdtr", DOMAIN );
return(0.0);
}
if( t == 0 )
return( 0.5 );
if( t < -2.0 )
{
rk = k;
z = rk / (rk + t * t);
p = 0.5 * incbet( 0.5*rk, 0.5, z );
return( p );
}
/* compute integral from -t to + t */
if( t < 0 )
x = -t;
else
x = t;
rk = k; /* degrees of freedom */
z = 1.0 + ( x * x )/rk;
/* test if k is odd or even */
if( (k & 1) != 0)
{
/* computation for odd k */
xsqk = x/sqrt(rk);
p = atan( xsqk );
if( k > 1 )
{
f = 1.0;
tz = 1.0;
j = 3;
while( (j<=(k-2)) && ( (tz/f) > MACHEP ) )
{
tz *= (j-1)/( z * j );
f += tz;
j += 2;
}
p += f * xsqk/z;
}
p *= 2.0/PI;
}
else
{
/* computation for even k */
f = 1.0;
tz = 1.0;
j = 2;
while( ( j <= (k-2) ) && ( (tz/f) > MACHEP ) )
{
tz *= (j - 1)/( z * j );
f += tz;
j += 2;
}
p = f * x/sqrt(z*rk);
}
/* common exit */
if( t < 0 )
p = -p; /* note destruction of relative accuracy */
p = 0.5 + 0.5 * p;
return(p);
}
double stdtri( k, p )
int k;
double p;
{
double t, rk, z;
int rflg;
if( k <= 0 || p <= 0.0 || p >= 1.0 )
{
mtherr( "stdtri", DOMAIN );
return(0.0);
}
rk = k;
if( p > 0.25 && p < 0.75 )
{
if( p == 0.5 )
return( 0.0 );
z = 1.0 - 2.0 * p;
z = incbi( 0.5, 0.5*rk, fabs(z) );
t = sqrt( rk*z/(1.0-z) );
if( p < 0.5 )
t = -t;
return( t );
}
rflg = -1;
if( p >= 0.5)
{
p = 1.0 - p;
rflg = 1;
}
z = incbi( 0.5*rk, 0.5, 2.0*p );
if( MAXNUM * z < rk )
return(rflg* MAXNUM);
t = sqrt( rk/z - rk );
return( rflg * t );
}
|
