Jeff Pitblado, in reply to Jeffrey Simons, writes (in part):
> There are several approximations to this (inverse non-central
> chi-square) distribution, most
> (if not all) of which are discussed in Johnson, Kotz, and
> Balakrishnan (1995). One approximation is based on a Central
> Limit Argument.
>
> Assume X2 is distributed as a non-central chi-square
> random variable
> with n degrees of freedome and non-centrality parameter L, then
>
> Z = (X2 - (n+L))/sqrt(2*(n+2*L))
>
> will tend to follow the standard normal distribution
> for fixed n and
> large L, or for fixed L and large n.
>
> I compared the results between -invnchi2()- and this
> approximation fixing n=200, using alpha = 0.05, and L ranging
> between 5 and 99. The relative difference between the two
> ranged between 0.0040 and 0.0047. The normal approximation
> was consistently less than the result from -invnchi2()-
> (minimum difference was 1.12, max was 1.38).
As Jeff says, there are a number of approximations. I repeated Jeff's
analysis with the 3 approximation formulas given in Abramowitz
& Stegun (1972) and show the results below. The first approximation,
labeled "invnchi2A", is based on the central chi-square while
approximations "invnchi2B" and "invnchi2C" are based on the central
normal distribution. I report Stata's invnchi2 value, the result of
each of these formulas and the relative difference of each from
invnchi2, using invnchi2 as the reference.
invnchi2A, based on the central chi-square, is very poor, averaging
5.037% relative difference and ranging from 0.064% to 11.831%. It was
lower than the Stata result throughout and became progressively more
divergent as lambda became larger.
invnchi2B is much better, at 0.139% average relative difference and
range 0.108% to 0.167%. It was slightly lower throughout than the
Stata value but became closer as lambda increased.
invnchi2C is quite good, at -0.020% average relative difference and
range -0.001% to -0.040%. It was slightly higher throughout than the
Stata value and became more divergent as lambda increased.
If I understood Jeff's relative difference notation, then the
approximation he used ranged in relative difference from 0.40% to 0.47%.
Approximation invnchi2C seems quite acceptable (for this example) and
could easily be programmed. It would be reasonable, though, to test
such an approximation formula over a wide range of values before
offering it as a stand-alone program (or, better yet, a function).
Tom Steichen
------------------------------------
local p = .05
local v = 200
forvalues l = 5/99 {
local a = `v' + `l'
local b = `l'/`a'
local stata = invnchi2(`v',`l',`p')
local invnchi2A = (1+`b') * invchi2(`v', `p')
local invnchi2B = ((1+`b')/2)*(invnorm(`p') + sqrt(2*`a'/(1+`b') - 1))^2
local invnchi2C = `a'*(invnorm(`p')*sqrt(2/9*((1+`b')/`a')) + 1 - 2/9*((1+`b')/`a'))^3
di %5.0f `l' %10.3f `stata' _c
di %10.3f `invnchi2A' %10.6f (`stata' - `invnchi2A')/`stata' _c
di %10.3f `invnchi2B' %10.6f (`stata' - `invnchi2B')/`stata' _c
di %10.3f `invnchi2C' %10.6f (`stata' - `invnchi2C')/`stata'
}
lambda invnchi2 invnchi2A rel_difA invnchi2B rel_difB invnchi2C rel_difC
5 172.494 172.383 0.000642 172.206 0.001666 172.495 -0.000009
6 173.340 173.180 0.000921 173.050 0.001668 173.340 -0.000005
7 174.185 173.969 0.001242 173.895 0.001667 174.187 -0.000006
8 175.031 174.751 0.001603 174.741 0.001661 175.033 -0.000011
9 175.880 175.525 0.002019 175.587 0.001668 175.881 -0.000004
10 176.726 176.292 0.002457 176.434 0.001654 176.729 -0.000017
11 177.575 177.051 0.002947 177.281 0.001653 177.578 -0.000018
12 178.423 177.804 0.003473 178.129 0.001648 178.427 -0.000022
13 179.272 178.549 0.004033 178.978 0.001639 179.277 -0.000030
14 180.124 179.287 0.004642 179.828 0.001642 180.128 -0.000026
15 180.972 180.019 0.005268 180.678 0.001626 180.980 -0.000041
16 181.824 180.744 0.005940 181.529 0.001622 181.832 -0.000043
17 182.678 181.462 0.006658 182.380 0.001630 182.684 -0.000035
18 183.529 182.173 0.007390 183.232 0.001619 183.537 -0.000044
19 184.384 182.878 0.008165 184.085 0.001620 184.391 -0.000042
20 185.235 183.577 0.008953 184.938 0.001603 185.246 -0.000058
21 186.092 184.269 0.009797 185.792 0.001612 186.101 -0.000047
22 186.943 184.955 0.010637 186.646 0.001589 186.956 -0.000069
23 187.800 185.635 0.011532 187.501 0.001592 187.812 -0.000064
24 188.657 186.308 0.012450 188.357 0.001592 188.669 -0.000062
25 189.514 186.976 0.013392 189.213 0.001590 189.526 -0.000064
26 190.371 187.638 0.014356 190.070 0.001584 190.384 -0.000068
27 191.228 188.294 0.015343 190.927 0.001575 191.242 -0.000075
28 192.085 188.944 0.016350 191.785 0.001564 192.101 -0.000084
29 192.942 189.589 0.017378 192.643 0.001550 192.961 -0.000097
30 193.802 190.228 0.018440 193.502 0.001547 193.820 -0.000097
31 194.661 190.861 0.019520 194.361 0.001542 194.681 -0.000101
32 195.521 191.489 0.020620 195.221 0.001534 195.542 -0.000107
33 196.381 192.112 0.021736 196.081 0.001524 196.403 -0.000115
34 197.243 192.729 0.022884 196.942 0.001525 197.265 -0.000112
35 198.105 193.341 0.024049 197.804 0.001524 198.128 -0.000112
36 198.965 193.948 0.025215 198.665 0.001506 198.990 -0.000127
37 199.828 194.550 0.026411 199.528 0.001500 199.854 -0.000132
38 200.693 195.147 0.027635 200.391 0.001505 200.718 -0.000125
39 201.555 195.738 0.028860 201.254 0.001494 201.582 -0.000133
40 202.418 196.325 0.030099 202.118 0.001481 202.447 -0.000145
41 203.280 196.907 0.031352 202.982 0.001465 203.312 -0.000158
42 204.148 197.484 0.032643 203.847 0.001474 204.178 -0.000147
43 205.010 198.056 0.033921 204.712 0.001455 205.044 -0.000164
44 205.878 198.624 0.035236 205.578 0.001459 205.911 -0.000157
45 206.743 199.187 0.036551 206.444 0.001449 206.778 -0.000166
46 207.609 199.745 0.037876 207.311 0.001436 207.645 -0.000176
47 208.477 200.299 0.039225 208.178 0.001434 208.513 -0.000176
48 209.344 200.849 0.040583 209.045 0.001431 209.382 -0.000177
49 210.212 201.394 0.041952 209.913 0.001425 210.250 -0.000181
50 211.080 201.934 0.043330 210.781 0.001417 211.120 -0.000186
51 211.948 202.471 0.044717 211.650 0.001408 211.989 -0.000193
52 212.816 203.003 0.046112 212.519 0.001396 212.859 -0.000203
53 213.684 203.531 0.047516 213.389 0.001383 213.730 -0.000214
54 214.555 204.054 0.048941 214.259 0.001380 214.601 -0.000214
55 215.425 204.574 0.050372 215.129 0.001376 215.472 -0.000216
56 216.296 205.089 0.051811 216.000 0.001370 216.344 -0.000219
57 217.167 205.601 0.053257 216.871 0.001362 217.216 -0.000225
58 218.037 206.109 0.054710 217.743 0.001352 218.088 -0.000232
59 218.908 206.612 0.056169 218.615 0.001341 218.961 -0.000241
60 219.781 207.112 0.057646 219.487 0.001340 219.834 -0.000239
61 220.655 207.608 0.059128 220.360 0.001338 220.708 -0.000239
62 221.528 208.100 0.060616 221.233 0.001334 221.582 -0.000241
63 222.402 208.589 0.062109 222.106 0.001328 222.456 -0.000244
64 223.275 209.073 0.063607 222.980 0.001321 223.331 -0.000249
65 224.149 209.554 0.065109 223.855 0.001312 224.206 -0.000255
66 225.022 210.032 0.066616 224.729 0.001301 225.081 -0.000263
67 225.895 210.506 0.068127 225.604 0.001289 225.957 -0.000273
68 226.769 210.976 0.069642 226.480 0.001275 226.833 -0.000284
69 227.648 211.443 0.071184 227.355 0.001285 227.710 -0.000272
70 228.521 211.906 0.072706 228.231 0.001268 228.587 -0.000286
71 229.397 212.366 0.074242 229.108 0.001262 229.464 -0.000290
72 230.273 212.823 0.075782 229.985 0.001254 230.341 -0.000295
73 231.152 213.276 0.077335 230.862 0.001257 231.219 -0.000289
74 232.026 213.726 0.078869 231.739 0.001235 232.097 -0.000309
75 232.905 214.173 0.080428 232.617 0.001235 232.976 -0.000306
76 233.784 214.616 0.081988 233.495 0.001234 233.855 -0.000305
77 234.662 215.056 0.083550 234.374 0.001231 234.734 -0.000305
78 235.539 215.493 0.085104 235.252 0.001215 235.614 -0.000319
79 236.418 215.927 0.086670 236.132 0.001210 236.494 -0.000322
80 237.296 216.358 0.088237 237.011 0.001203 237.374 -0.000326
81 238.178 216.786 0.089816 237.891 0.001206 238.254 -0.000320
82 239.057 217.211 0.091386 238.771 0.001196 239.135 -0.000327
83 239.936 217.632 0.092956 239.651 0.001186 240.016 -0.000335
84 240.815 218.051 0.094528 240.532 0.001174 240.898 -0.000344
85 241.696 218.467 0.096111 241.413 0.001172 241.779 -0.000343
86 242.578 218.880 0.097694 242.295 0.001169 242.662 -0.000344
87 243.457 219.290 0.099267 243.176 0.001153 243.544 -0.000357
88 244.341 219.697 0.100861 244.058 0.001159 244.427 -0.000349
89 245.220 220.101 0.102434 244.941 0.001141 245.310 -0.000364
90 246.105 220.503 0.104028 245.823 0.001144 246.193 -0.000358
91 246.984 220.902 0.105602 246.706 0.001124 247.076 -0.000376
92 247.868 221.298 0.107195 247.589 0.001125 247.960 -0.000372
93 248.752 221.691 0.108788 248.473 0.001124 248.844 -0.000370
94 249.637 222.082 0.110380 249.357 0.001123 249.729 -0.000369
95 250.518 222.470 0.111962 250.241 0.001109 250.614 -0.000380
96 251.400 222.855 0.113543 251.125 0.001095 251.499 -0.000392
97 252.285 223.238 0.115133 252.010 0.001090 252.384 -0.000394
98 253.169 223.618 0.116722 252.895 0.001084 253.269 -0.000397
99 254.053 223.996 0.118310 253.780 0.001077 254.155 -0.000401
Averages:
212.971 201.387 5.037% 212.678 0.139% 213.016 -0.020%
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