A Course in Mathematical Statistics phần 10 pps

The proof of the theorems formulated in this appendix may be found in most books of linear algebra.. Murdoch, Linear Algebra for Undergraduates, Wiley, 1957; S.. Rao, Linear Statistical

rank rank rank A1+ A2+ ( I A − 1− A2) = n.

iv) If Aj, j = 1, , m are symmetric idempotent matrices of the same order

and Σm

j=1Aj is also idempotent, the AiAj= 0 for 1 ≤ i < j ≤ m.

The proof of the theorems formulated in this appendix may be found in

most books of linear algebra For example, see Birkhoff and MacLane, A

Survey of Modern Algebra, 3d ed., MacMillan, 1965; S Lang, Linear Algebra,

Addison-Wesley, 1968; D C Murdoch, Linear Algebra for Undergraduates, Wiley, 1957; S Perlis, Theory of Matrices, Addison-Wesley, 1952 For a brief

exposition of most results from linear algebra employed in statistics, see also

C R Rao, Linear Statistical Inference and Its Applications, Chapter 1, Wiley, 1965; H Scheffé, The Analysis of Variance, Appendices I and II, Wiley, 1959; and F A Graybill, An Introduction to Linear Statistical Models, Vol I, Chap-

ter 1, McGraw-Hill, 1961.

II.1 Noncentral t-Distribution

It was seen in Chapter 9, Application 2, that if the independent r.v.’s X and Y were distributed as N(0, 1) and χ2

r, respectively, then the distribution of the r.v.

T = X ( ) Y r was the (Student’s) t-distribution with r d.f Now let X and

r, respectively, and set

T X Y r The r.v T ′ is said to have the noncentral t-distribution with r

this distribution, if often denoted by t ′r:δ Using the definition of a t ′r:δ r.v., it can

be found by well known methods that its p.d.f is given by

1

1 2

1 2

1 2 0

chi-square distribution with r d.f and noncentrality parameter δ, where δ2

r:δ Using the definition of a χ′2

r;δ r.v., one can find its p.d.f but it does not have any simple closed form It can be seen that this p.d.f is a mixture

22

0

! and is the p.d.f of , , 1, .

II.3 Noncentral F-Distribution

In Chapter 9, Application 2, the F-distribution with r1 and r2 d.f was defined as the distribution of the r.v.

.

Then the distribution of F ′ is said to be the noncentral F-distribution with r1and

r2d.f and noncentrality parameter δ This distribution, and also an r.v having

this distribution, is often denoted by F ′r

1 112

1 2 1 2

1 2

0 1

1 2

, , ,

REMARKS(i) By setting δ = 0 in the noncentral t, χ2

and F-distributions, we obtain the

t, χ2

and F-distributions, respectively In view of this, the latter tions may also be called central t, χ2

and F-distributions.

(ii) Tables for the noncentral t, χ2

and F-distributions are given in a reference cited elsewhere, namely, Handbook of Statistical Tables by D B Owen.

Addison-Wesley, 1962.

Appendix III

Tables

Table 1 The Cumulative Binomial Distribution

The tabulated quantity is

Table 2 The Cumulative Poisson DistributionThe tabulated quantity is

j

e j

=

−

∑0

Table 3 The Normal Distribution

The tabulated quantity is

Table 4 Critical Values for Student’s t-Distribution

freedom Then the tabulated quantities are the numbers x for which

Table 5 Critical Values for the Chi-Square Distribution

Let χ2

freedom Then the tabulated quantities are the numbers x for which

Let F r1,r2 be a random variable having the F-distribution with r1, r2 degrees of freedom.

Then the tabulated quantities are the numbers x for which

1 0.950 161.45 199.50 215.71 224.58 230.16 233.99 0.950 10.975 647.79 799.50 864.16 899.58 921.85 937.11 0.9750.990 4052.2 4999.5 5403.3 5624.6 5763.7 5859.0 0.9900.995 16211 20000 21615 22500 23056 23437 0.9950.500 0.66667 1.0000 1.1349 1.2071 1.2519 1.2824 0.5000.750 2.5714 3.0000 3.1534 3.2320 3.2799 3.3121 0.7500.900 8.5623 9.0000 9.1618 9.2434 9.2926 9.3255 0.900

2 0.950 18.513 19.000 19.164 19.247 19.296 19.330 0.950 20.975 38.506 39.000 39.165 39.248 39.298 39.331 0.9750.990 98.503 99.000 99.166 99.249 99.299 99.332 0.9900.995 198.50 199.00 199.17 199.25 199.30 199.33 0.9950.500 0.58506 0.88110 1.0000 1.0632 1.1024 1.1289 0.5000.750 2.0239 2.2798 2.3555 2.3901 2.4095 2.4218 0.7500.900 5.5383 5.4624 5.3908 5.3427 5.3092 5.2847 0.900

3 0.950 10.128 9.5521 9.2766 9.1172 9.0135 8.9406 0.950 30.975 17.443 16.044 15.439 15.101 14.885 14.735 0.9750.990 34.116 30.817 29.457 28.710 28.237 27.911 0.990

r2 0.995 55.552 49.799 47.467 46.195 45.392 44.838 0.995 r2

0.500 0.54863 0.82843 0.94054 1.0000 1.0367 1.0617 0.5000.750 1.8074 2.0000 2.0467 2.0642 2.0723 2.0766 0.7500.900 4.5448 4.3246 4.1908 4.1073 4.0506 4.0098 0.900

4 0.950 7.7086 6.9443 6.5914 6.3883 6.2560 6.1631 0.950 40.975 12.218 10.649 9.9792 9.6045 9.3645 9.1973 0.9750.990 21.198 18.000 16.694 15.977 15.522 15.207 0.9900.995 31.333 26.284 24.259 23.155 22.456 21.975 0.9950.500 0.52807 0.79877 0.90715 0.96456 1.0000 1.0240 0.5000.750 1.6925 1.8528 1.8843 1.8927 1.8947 1.8945 0.7500.900 4.0604 3.7797 3.6195 3.5202 3.4530 3.4045 0.900

5 0.950 6.6079 5.7861 5.4095 5.1922 5.0503 4.9503 0.950 50.975 10.007 8.4336 7.7636 7.3879 7.1464 6.9777 0.9750.990 16.258 13.274 12.060 11.392 10.967 10.672 0.9900.995 22.785 18.314 16.530 15.556 14.940 14.513 0.9950.500 0.51489 0.77976 0.88578 0.94191 0.97654 1.0000 0.5000.750 1.6214 1.7622 1.7844 1.7872 1.7852 1.7821 0.7500.900 3.7760 3.4633 3.2888 3.1808 3.1075 3.0546 0.900

6 0.950 5.9874 5.1433 4.7571 4.5337 4.3874 4.2839 0.950 60.975 8.8131 7.2598 6.5988 6.2272 5.9876 5.8197 0.9750.990 13.745 10.925 9.7795 9.1483 8.7459 8.4661 0.9900.995 18.635 14.544 12.917 12.028 11.464 11.073 0.995

0.990 8.2600 8.1016 7.9761 7.8741 7.7891 7.7183 0.990

0.995 10.786 10.566 10.391 10.250 10.132 10.034 0.995

Table 6 (continued )

r1

0.500 2.0773 2.0858 2.0931 2.1104 2.1190 2.1321 0.5000.750 9.4399 9.4685 9.4934 9.5520 9.5813 9.6255 0.7500.900 60.903 61.073 61.220 61.567 61.740 62.002 0.900

1 0.950 244.69 245.37 245.95 247.32 248.01 249.05 0.950 10.975 979.85 982.54 984.87 990.36 993.10 997.25 0.9750.990 6125.9 6142.7 6157.3 6191.6 6208.7 6234.6 0.9900.995 24504 24572 24630 24767 24836 24940 0.9950.500 1.3672 1.3725 1.3771 1.3879 1.3933 1.4014 0.5000.750 3.3997 3.4051 3.4098 3.4208 3.4263 3.4345 0.7500.900 9.4145 9.4200 9.4247 9.4358 9.4413 9.4496 0.900

2 0.950 19.419 19.424 19.429 19.440 19.446 19.454 0.950 20.975 39.421 39.426 39.431 39.442 39.448 39.456 0.9750.990 99.422 99.427 99.432 99.443 99.449 99.458 0.9900.995 199.42 199.43 199.43 199.44 199.45 199.46 0.9950.500 1.2025 1.2071 1.2111 1.2205 1.2252 1.2322 0.5000.750 2.4520 2.4537 2.4552 2.4585 2.4602 2.4626 0.7500.900 5.2097 5.2047 5.2003 5.1898 5.1845 5.1764 0.900

3 0.950 8.7286 8.7148 8.7029 8.6744 8.6602 8.6385 0.940 30.975 14.305 14.277 14.253 14.196 14.167 14.124 0.9750.990 26.983 26.923 26.872 26.751 26.690 26.598 0.990

r2 0.995 43.271 43.171 43.085 42.880 42.778 42.622 0.955 r2

0.500 1.1305 1.1349 1.1386 1.1473 1.1517 1.1583 0.5000.750 2.0827 2.0828 2.0829 2.0828 2.0828 2.0827 0.7500.900 3.8853 3.8765 3.8689 3.8525 3.8443 3.8310 0.900

4 0.950 5.8910 5.8732 5.8578 5.8209 5.8025 5.7744 0.950 40.975 8.7148 8.6836 8.6565 8.5921 8.5599 8.5109 0.9750.990 14.306 14.248 14.198 14.079 14.020 13.929 0.9900.995 20.602 20.514 20.438 20.257 20.167 20.030 0.9950.500 1.0903 1.0944 1.0980 1.1064 1.1106 1.1170 0.5000.750 1.8867 1.8858 1.8851 1.8830 1.8820 1.8802 0.7500.900 3.2566 3.2466 3.2380 3.2171 3.2067 3.1905 0.900

5 0.950 4.6550 4.6356 4.6188 4.5783 4.5581 4.5272 0.950 50.975 6.4873 6.4554 6.4277 6.3616 6.3285 6.2780 0.9750.990 9.8244 9.7697 9.7222 9.6092 9.5527 9.4665 0.9900.995 13.292 13.214 13.146 12.984 12.903 12.780 0.9950.500 1.0647 1.0687 1.0722 1.0804 1.0845 1.0907 0.5000.750 1.7650 1.7634 1.7621 1.7586 1.7569 1.7540 0.7500.900 2.8918 2.8808 2.8712 2.8479 2.8363 2.8183 0.900

6 0.950 3.9761 3.9558 3.9381 3.8955 3.8742 3.8415 0.950 60.975 5.3287 5.2966 5.2687 5.2018 5.1684 5.1172 0.9750.990 7.6570 7.6045 7.5590 7.4502 7.3958 7.3127 0.9900.995 9.9494 9.8769 9.8140 9.6639 9.5888 9.4741 0.995

0.990 7.2285 7.1432 7.1000 7.0568 6.9690 6.8801 0.990

0.995 9.3583 9.2408 9.1814 9.1219 9.0015 8.8793 0.995

Table 6 (continued )

r1

0.500 0.50572 0.76655 0.87095 0.92619 0.96026 0.98334 0.5000.750 1.5732 1.7010 1.7169 1.7157 1.7111 1.7059 0.7500.900 3.5894 3.2574 3.0741 2.9605 2.8833 2.8274 0.900

7 0.950 5.5914 4.7374 4.3468 4.1203 3.9715 3.8660 0.950 70.975 8.0727 6.5415 5.8898 5.5226 5.2852 5.1186 0.9750.990 12.246 9.5466 8.4513 7.8467 7.4604 7.1914 0.9900.995 16.236 12.404 10.882 10.050 9.5221 9.1554 0.9950.500 0.49898 0.75683 0.86004 0.91464 0.94831 0.97111 0.5000.750 1.5384 1.6569 1.6683 1.6642 1.6575 1.6508 0.7500.900 3.4579 3.1131 2.9238 2.8064 2.7265 2.6683 0.900

8 0.950 5.3177 4.4590 4.0662 3.8378 3.6875 3.5806 0.950 80.975 7.5709 6.0595 5.4160 5.0526 4.8173 4.6517 0.9750.990 11.259 8.6491 7.5910 7.0060 6.6318 6.3707 0.9900.995 14.688 11.042 9.5965 8.8051 8.3018 7.9520 0.9950.500 0.49382 0.74938 0.85168 0.90580 0.93916 0.96175 0.5000.750 1.5121 1.6236 1.6315 1.6253 1.6170 1.6091 0.7500.900 3.3603 3.0065 2.8129 2.6927 2.6106 2.5509 0.900

9 0.950 5.1174 4.2565 3.8626 3.6331 3.4817 3.3738 0.950 90.975 7.2093 5.7147 5.0781 4.7181 4.4844 4.3197 0.9750.990 10.561 8.0215 6.9919 6.4221 6.0569 5.8018 0.990

r2 0.995 13.614 10.107 8.7171 7.9559 7.4711 7.1338 0.995 r2

0.500 0.48973 0.74349 0.84508 0.89882 0.93193 0.95436 0.5000.750 1.4915 1.5975 1.6028 1.5949 1.5853 1.5765 0.7500.900 3.2850 2.9245 2.7277 2.6053 2.5216 2.4606 0.900

10 0.950 4.9646 4.1028 3.7083 3.4780 3.3258 3.2172 0.950 100.975 6.9367 5.4564 4.8256 4.4683 4.2361 4.0721 0.9750.990 10.044 7.5594 6.5523 5.9943 5.6363 5.3858 0.9900.995 12.826 9.4270 8.0807 7.3428 6.8723 6.5446 0.9950.500 0.48644 0.73872 0.83973 0.89316 0.92608 0.94837 0.5000.750 1.4749 1.5767 1.5798 1.5704 1.5598 1.5502 0.7500.900 3.2252 2.8595 2.6602 2.5362 2.4512 2.3891 0.900

11 0.950 4.8443 3.9823 3.5874 3.3567 3.2039 3.0946 0.950 110.975 6.7241 5.2559 4.6300 4.2751 4.0440 3.8807 0.9750.990 9.6460 7.2057 6.2167 5.6683 5.3160 5.0692 0.9900.995 12.226 8.9122 7.6004 6.8809 6.4217 6.1015 0.9950.500 0.48369 0.73477 0.83530 0.88848 0.92124 0.94342 0.5000.750 1.4613 1.5595 1.5609 1.5503 1.5389 1.5286 0.7500.900 3.1765 2.8068 2.6055 2.4801 2.3940 2.3310 0.900

12 0.950 4.7472 3.8853 3.4903 3.2592 3.1059 2.9961 0.950 120.975 6.5538 5.0959 4.4742 4.1212 3.8911 3.7283 0.9750.990 9.3302 6.9266 5.9526 5.4119 5.0643 4.8206 0.9900.995 11.754 8.5096 7.2258 6.5211 6.0711 5.7570 0.995

0.990 4.6395 4.4994 4.3875 4.2961 4.2193 4.1553 0.990

0.995 5.5245 5.3451 5.2021 5.0855 4.9878 4.9063 0.995

Table 6 (continued )

r1

0.500 1.0469 1.0509 1.0543 1.0624 1.0664 1.0724 0.5000.750 1.6819 1.6799 1.6781 1.6735 1.6712 1.6675 0.7500.900 2.6543 2.6425 2.6322 2.6072 2.5947 2.5753 0.900

7 0.950 3.5501 3.5291 3.5108 3.4666 3.4445 3.4105 0.950 70.975 4.6281 4.5958 4.5678 4.5004 4.4667 4.4150 0.9750.990 6.4096 6.3585 6.3143 6.2084 6.1554 6.0743 0.9900.995 8.0962 8.0274 7.9678 7.8253 7.7540 7.6450 0.9950.500 1.0339 1.0378 1.0412 1.0491 1.0531 1.0591 0.5000.750 1.6216 1.6191 1.6170 1.6115 1.6088 1.6043 0.7500.900 2.4875 2.4750 2.4642 2.4378 2.4246 2.4041 0.900

8 0.950 3.2588 3.2371 3.2184 3.1730 3.1503 3.1152 0.950 80.975 4.1618 4.1293 4.1012 4.0334 3.9995 3.9472 0.9750.990 5.6085 5.5584 5.5151 5.4111 5.3591 5.2793 0.9900.995 6.9377 6.8716 6.8143 6.6769 6.6082 6.5029 0.9950.500 1.0239 1.0278 1.0311 1.0390 1.0429 1.0489 0.5000.750 1.5756 1.5729 1.5705 1.5642 1.5611 1.5560 0.7500.900 2.3638 2.3508 2.3396 2.3121 2.9893 2.2768 0.900

9 0.950 3.0472 3.0252 3.0061 2.9597 2.9365 2.9005 0.950 90.975 3.8302 3.7976 3.7694 3.7011 3.6669 3.6142 0.9750.990 5.0540 5.0048 4.9621 4.8594 4.8080 4.7290 0.990

r2 0.995 6.1524 6.0882 6.0325 5.8987 5.8318 5.7292 0.995 r2

0.500 1.0161 1.0199 1.0232 1.0310 1.0349 1.0408 0.5000.750 1.5395 1.5364 1.5338 1.5269 1.5235 1.5179 0.7500.900 2.2685 2.2551 2.2435 2.2150 2.2007 2.1784 0.900

10 0.950 2.8868 2.8644 2.8450 2.7977 2.7740 2.7372 0.950 100.975 3.5827 3.5500 3.5217 3.4530 3.4186 3.3654 0.9750.990 4.6491 4.6004 4.5582 4.4563 4.4054 4.3269 0.9900.995 5.5880 5.5252 5.4707 5.3396 5.2740 5.1732 0.9950.500 1.0097 1.0135 1.0168 1.0245 1.0284 1.0343 0.5000.750 1.5102 1.5069 1.5041 1.4967 1.4930 1.4869 0.7500.900 2.1927 2.1790 2.1671 2.1377 2.1230 2.1000 0.900

11 0.950 2.7611 2.7383 2.7186 2.6705 2.6464 2.6090 0.950 110.975 3.3913 3.3584 3.3299 3.2607 3.2261 3.1725 0.9750.990 4.3411 4.2928 4.2509 4.1496 4.0990 4.0209 0.9900.995 5.1642 5.1024 5.0489 4.9198 4.8552 4.7557 0.9950.500 1.0044 1.0082 1.0115 1.0192 1.0231 1.0289 0.5000.750 1.4861 1.4826 1.4796 1.4717 1.4678 1.4613 0.7500.900 2.1311 2.1170 1.1049 2.0748 2.0597 2.0360 0.900

12 0.950 2.6598 2.6368 2.6169 2.5680 2.5436 2.5055 0.950 120.975 3.2388 3.2058 3.1772 3.1076 3.0728 3.0187 0.9750.990 4.0993 4.0512 4.0096 3.9088 3.8584 3.7805 0.9900.995 4.8352 4.7742 4.7214 4.5937 4.5299 4.4315 0.995

0.990 3.7008 3.6192 3.5774 3.5355 3.4494 3.3608 0.990

0.995 4.3309 4.2282 4.1756 4.1229 4.0149 3.9039 0.995

Table 6 (continued )

r1

0.500 0.48141 0.73145 0.83159 0.88454 0.91718 0.93926 0.5000.750 1.4500 1.5452 1.5451 1.5336 1.5214 1.5105 0.7500.900 3.1362 2.7632 2.5603 2.4337 2.3467 2.2830 0.900

13 0.950 4.6672 3.8056 3.4105 3.1791 3.0254 2.9153 0.950 130.975 6.4143 4.9653 4.3472 3.9959 3.7667 3.6043 0.9750.990 9.0738 6.7010 5.7394 5.2053 4.8616 4.6204 0.9900.995 11.374 8.1865 6.9257 6.2335 5.7910 5.4819 0.9950.500 0.47944 0.72862 0.82842 0.88119 0.91371 0.93573 0.5000.750 1.4403 1.5331 1.5317 1.5194 1.5066 1.4952 0.7500.900 3.1022 2.7265 2.5222 2.3947 2.3069 2.2426 0.900

14 0.950 4.6001 3.7389 3.3439 3.1122 2.9582 2.8477 0.950 140.975 6.2979 4.8567 4.2417 3.8919 3.6634 3.5014 0.9750.990 8.8616 6.5149 5.5639 5.0354 4.6950 4.4558 0.9900.995 11.060 7.9216 6.6803 5.9984 5.5623 5.2574 0.9950.500 0.47775 0.72619 0.82569 0.87830 0.91073 0.93267 0.5000.750 1.4321 1.5227 1.5202 1.5071 1.4938 1.4820 0.7500.900 3.0732 2.6952 2.4898 2.3614 2.2730 2.2081 0.900

15 0.950 4.5431 3.6823 3.2874 3.0556 2.9013 2.7905 0.950 150.975 6.1995 4.7650 4.1528 3.8043 3.5764 3.4147 0.9750.990 8.6831 6.3589 5.4170 4.8932 4.5556 4.3183 0.990

r2 0.995 10.798 7.7008 6.4760 5.8029 5.3721 5.0708 0.995 r2

0.500 0.47628 0.72406 0.82330 0.87578 0.90812 0.93001 0.5000.750 1.4249 1.5137 1.5103 1.4965 1.4827 1.4705 0.7500.900 3.0481 2.6682 2.4618 2.3327 2.2438 2.1783 0.900

16 0.950 4.4940 3.6337 3.2389 3.0069 2.8524 2.7413 0.950 160.975 6.1151 4.6867 4.0768 3.7294 3.5021 3.3406 0.9750.990 8.5310 6.2262 5.2922 4.7726 4.4374 4.2016 0.9900.995 10.575 7.5138 6.3034 5.6378 5.2117 4.9134 0.9950.500 0.47499 0.72219 0.82121 0.87357 0.90584 0.92767 0.5000.750 1.4186 1.5057 1.5015 1.4873 1.4730 1.4605 0.7500.900 3.0262 2.6446 2.4374 2.3077 2.2183 2.1524 0.900

17 0.950 4.4513 3.5915 3.1968 2.9647 2.8100 2.6987 0.950 170.975 6.0420 4.6189 4.0112 3.6648 3.4379 3.2767 0.9750.990 8.3997 6.1121 5.1850 4.6690 4.3359 4.1015 0.9900.995 10.384 7.3536 6.1556 5.4967 5.0746 5.7789 0.9950.500 0.47385 0.72053 0.81936 0.87161 0.90381 0.92560 0.5000.750 1.4130 1.4988 1.4938 1.4790 1.4644 1.4516 0.7500.900 3.0070 2.6239 2.4160 2.2858 2.1958 1.1296 0.900

18 0.950 4.4139 3.5546 3.1599 2.9277 2.7729 2.6613 0.950 180.975 5.9781 4.5597 3.9539 3.6083 3.3820 3.2209 0.9750.990 8.2854 6.0129 5.0919 4.5790 4.2479 4.0146 0.9900.995 10.218 7.2148 6.0277 5.3746 4.9560 4.6627 0.995

0.990 3.8406 3.7054 3.5971 3.5082 3.4331 3.3706 0.990

0.995 4.4448 4.2759 4.1410 4.0305 3.9374 3.8599 0.995

These tables have been adapted from Donald B Owen’s Handbook of Statistical Tables,

published by Addison-Wesley, by permission of the publishers