NBS PROJECT NBS REPORT

20 September 1951

:i!03°21-5X06 1X67

TABLE OF THE FIRST MOMENT OF RANKED EXTREMES

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

Supported by the

National Advisory Committee

for Aeronautics

This report is issued for

in any form, either in who!

fro* the Office of the Dire

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

reprinting, or reproduction

ision in writing is obtained

ngton 25, D. C.

FOREWORD

The tabulation of the first moment of ranked extremes

will considerably simplify the use of certain methods of ana«

Xyzing extreme-value data 0 The present table was computed by

the Computation Laboratory of the National Applied Mathematics

Laboratories, National Bureau of Standards, at the suggestion

of Bradford F„ Kimball , who developed the necessary formula®

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> This table is one item of a project aimed at the improved

application of the theory of extreme values to the analysis of

gust loads of airplanes 0 This program of the Statistical

Engineering Laboratory (Section 3 of the National Applied

Mathematic® Laboratories, Division 11, NBS) ie supported by

the National Advisory Committee for Aeronautics 0

tT. Ho Curtiss

Chiof 9 National Applied

Mathematics Laboratories

So Ua Condon

Director

National Bureau of Standards

20 September 1951
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TABLE OF THE FIRST MOMENT OF RANKED EXTREMES

Shit table gives the expected values of order statistics

from the distribution of largest values Frob {Y< y}TM ©xpt®*©^} 0 / The entries were computed from the formula given by BoFo Kimball

B(yP ) * 0 2 9

b&G

where y0 > 7i > *°o yr

> ooo©*oc £ yn

^ denote the n order

statistics in a sample of n values and 0 is Euler f s constant

( 05772^)0 The expected values of y^* E/fyp } e are given in the

table for

r ~ 0(l)n~l or 25* shiehover Is smaller

) n - 1(1)10(5)60(10)100©

The table has two important uses, One* described by

Kimball (ibid) e is in connection with a method of graphical

analysis of an ordered sample of extremes*

A second use relates to determination of the w b®stw linear

unbiased estimator T of the mode u of the distribution of

largest values* Prob {X < x) « ®xp(^G°J) * where y

** a(x ® u)*

regardless of whether or not the parameter a i & known 0 If

^Kimball* B© F©2 Assignment of Frequencies to a Completely

Ordered Set of Sample Data* Trans© Amer© Geophysical Union*

volo 27 * no© 6* X94&* Pp* oi(.3

®6l|.6 © See also discussion of

this article in vol© 26 * no© 6* 194? * PP° 951-953 °
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T 55 2 fjZ-* 9 whore the x* are n ordered sample values (order

1*0 1 * • " *•••

statistics) xQ > xx > ooo > ^rom this distribution, than

on© set of conditions on the unknowns is given by the vanish^

ing of the bias of T, that is, E{T) ® u * 0 identically in u©

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' •* * ' ' /, V% V i From the above linear relation between x and j s w© have

E(y^) * a[E(x^) « u] # whence

whose vanishing requires that the satisfy the two conditions

most efficient linear estimator T (i©© 0 , the one with smallest

variance), the must be determined by further conditions' so

B(T) - u * 2 <=> u * 2Tj;[u * ~ E(y^) J - us

1, X^iEfyj) * 0© In order to determine the-^best* or
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• c + 2i (-l) t

(t)i6k c Xn (n-t) „ where

'X* o

Constant - 0.5773157 ,

C Bul« 1 S

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3* m

n^X n»*2 ns*3 3? 0.57721' 566 +1.2703 628 1,6758- 280 0

1

2

-0.1159 315 +0.4594 326 , -0,4036 136

12

r n«»4 n“»5 .n**6 T

0 1.9655 100 2 . 1866 536 2.3689 75X 0

x.. 0.8127 317 ! 1.0709 358 1,2750 458 1

2 +0.1060 835 +0 ,4255 596 0,6627 159. 2

3 -0,5735 126 -0.1068 945 +O.X885 853

4 -0.6901 6?1 -0.2545 345

5 -0,7772 937 5

p n*®7 n«8 n«*9 p

0 2.5231 253 2.6566 572 • 2.7744 402 0

1 1.4440 711 1.5884 c6l 1.7143 929 X

o C.8524 826 1.0110 660 1.14?4 521 2

3 0.4096 935 0.5881 770 0.7382 939 3

4 +0.0224 042 +0.23I2 101 0.4005 308 4

5 -0.3653 '099 , -0.1028 793 +0.0957 524 3

6 -0.8459 576 -0.4527 868 -0.2021 957 6

7 -0.9021 249 -0.5243 843 7

8 --0.9493 425 8

r 12“ 10 n-’15 n«s2 0 p

0 2.8798 008 3.2852 659 3.5729 4?9 f

*\ 0

X I.826I 956 2.2503 ”28 2.5470 821 x

2 1.2671 822 1.7132 872 2.0200 359 2

3 0 . 8680 83.8 1.3403 534 ' 1.6583 531 3

4 0.5436 122 1,0478 353 1 ,3705 557 4

5 +0.2574 495 0.8013 873

0,5851 754

1.1471 396 5

6 -0.0120 439 0.9472 328 6

ml -O.2836 893 0.3872 821 0.7690 6?1 7-C.5845 581 0.2010 161 O.6063 833 8

.9 -0.9898 741 +0,0206 140 0.4548 34? 9

10 -0.1594 579 0 .3111 607 10

XX -0.3458 052 0.1727 124 11

12 V -0.5485 327. +0.0371 321 12

13 -0.7883 747 -0.0979 135 13

14

X3 , 16

17

-1.1326 841 -0.2350 333

-0,3775 607

-0.5304 220

-0.7022 321

.14

1 PC 16

17

18 -O.9II9 529 18

19 -1.2232 104 IQ

\7l r j ss C * X (-1)" 1 1,) A vhere C **

«£ 3 £, Constant ® c. 577215?

r n«?5 n«*30 n*35 r

0 3.7960 915 3-9784 130 4.1325 537 0

1 2.7755 4l6 2.9613 665 3.1179 999 1

2.2 C 42 557 2.4438 174 2.6030 713 2

• ; 1.8988 407 2.0923 991 2.2544 241 3

4 1.6258 959 1.8237 416 1.9887 029 4

5 1.5020 114 1.6044 708 1.7725 500 5

6 1 2104 327 1.4178 745 1.5892 723 6

7 1.0415 346 1.2543 779 1.4293 167 7

8 r.8892 472 1.1079 713 1.2866 988 8

9 J.7494 644 0.9746 216 1.1574 14? 9

10 0.6192 453 0.8514 772 1.0386 471 10

11 0.4963 746 0.7364 333 0.9283 319 11

12 0.3791 010 0.6278 773 0.8249 044 12

13 0.2659 691 0.5245 305 0.7271 437 13

14 0.1557 008 0.4253 445 0.6340 +12

0.5448 828

14

15 +0.0471 025 0.3294 297 13

16 -0.0610 213 0.2360 031 0.4589 015 16

r -0.1699 655 0.1443 471 0.3755 427 17

3i -0.2812 545 +0.0537 740 0.2942 883 18

‘9 .

' -0.3968 504 -0.0364 087 0.2146 659 19

20 -0.5195 530 -0.1269 375 0.1362 310 20

21 -0.6536 981 -0.2186 413 +0.0585 513 21

22 -0.8073 485 -0.3125 115 -0.0183 098 22

23 -0.9984 861 -0.4098 117 -0.0963 no 23

24 -1.2882 598 -0.5122 664 -0.1744 547 24

23 -0.6224 196 -0.2538 157 25

n**4o n»45 n«50 r

0 4.2660 951 4.3838 782 4.4892 387 0

1 3-2533 828 3.3725 996 3.4791 033 1

2 2.7403 936 2.8611 039 2.9687 929 2

3 2.3937 809 2.5160 483 2.6249 676 3

4 2.1301 969 2.2540 897 2.3642 868 k

3 1.9162 926 2.0418 841 2.1534 099 5

6 1.7353 848 1.8627 537 1.9756 620 6

7 1.5779 311 1.7071 623 1.8215 109 T

8 1.4379 596 1.5691 446 1.6849 953 8

1.31X4 8c4 1.4447 183 1.5621 370 9

1.1956 921 1.3310 904 1.4501 482 10

11 I.0885 487 1.2262 243 1.3459 974 11

12 0.9885 067 1,1285 871 1.2511 576 * 12

13 0.8943 693 1.0369 939 1.1614 507 13

14 0,8051 863 0.9505 080 1,0769 4-70 1H-

, 13 0.72.01 865 0.8683 738 0.9968 991 15

lo 0.6387 319

‘ 0.7899 709 0.9206 958 16

17 0,5602 845 0.7147 813 O.S478 294 IT

18 0.4843 816 0 6423 661 • 0.7778 727 18

19 0.;4106 182 0.5723 4?6 0,7104 609 19

70 0.3386 325 0.5043 959 0.6452 792 20

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r

») «* 0 * it) •& " Xw (n-t), where C ** Euler's

v Constant 0.577 'X57

'fx'mhO rs-4.5

0 .2630 942

0.1986 951

0.3301 405

+0.0621 4oS

-0.0055 96?

0.4>82 136

0.3 35 525

0.3 .01 567

0.2 78 ®6$

0.1162 904

0.5820 524 . 21

0.5205 370 22

0.4605 143 2?

0.4017 883 24

0,3441 759 25

r n*«60 ja-70 r

0 4.5845 489 4 . 6;’ 15 602 4.8257 109 0

1 3,5753 462 3 * & 31 331 J. 8184 993 x

2 3.0659 996 3 * X

: 45 855 3.3111 996 i

3 2.7231 708 O £! .28 80S 2,9704 763 c

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4 2.4635 211 2.5 37 800 2.7129 934 si 5 2 .2837 117 2„> 48 473 ' 2 ,5054 151 3

6 2 0770 701 2 , 11:91 110 2.3310 '720 0

T 1.9240 664 2 oC .70 428 2.1804 376

,-**7 r

. ) 8 1.7887 415 1.8; 2o 854 2.0475 566 8

9 1.6671 204 XT’ ,20 653 1 .9284 576- 9

xb 1.5564 180 1.61 •23 995 1 .8203 598 10

11 1 1.4546 060 1.5! ;X6 619 1.7212 39-4 11

12 I.. 360I 612 1

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,83 314 I .6295 ?8o 12

13 1,2719 091 1.3 v 12 36O 1.5442 063 13

14 1,1889 245 1.2 ?94 532 1.4642 047 14

X

i3 1.1104 647 X .2 .22 431 1,3888 36? 13

16 1.0359 236 X .X •>90 029 1.3175 026 16

XT 0.964-7 993 x'.b : 92 342 1.2497 0?8 I?

.18 0.8966 70S 1.0 )25 197 1.1350 392 IS

19 0.8311 80? 0.9 •85 062 1.1231 476 19

20 0.7530 221 0,8 68 913 1 . 0637 "SoS 20

21 0.7069 285 0.8 .74 137 I.OO65 469 21

22 0.6476 664 0 .7 >98 436 0.9513 392 22

2;> 0.5900 293 0 .7 >39 865 0.8979 781 23

24 0.5338 321 0.6496 534 0.8462 324 — ? 25 0.4789 079 0.5 67 022 0.7959 700 25

P nw80 bf-«’90 ! 3»100 - r

0 4.9592 423 5.0 TO 263 5,1823 859 0

x 3.9529 337 4 . a

; 14 283 4.1773 323 X

2 3.4465 TOC 3.5 r>7 ?84 3.6722 760 0£. 3 3.1067 984 5} >67 412 3.3338 229 3

4 2.8502 897 2 . 9 r09 823 3.0786 589 5

5 2.6437 091 «

>• 551 672 2.8734 497 5

6 2.4703 8Sl 2,5 926 280 2.7015 279 6

7 2.3208 Oil 2 .

*:• 38 397 2.5533 687 7

8 2. 1889 338 p ^ 1.23 486 2,4230 189 8

9 2.0709 961 td q , -

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956 850 2.3065 092 0•cP '7 n 1.964b 281 2.0 B95 700 2.2010 610 1C

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*» C -f 2. ( -X) * iJ ,6^ (n-t) . vh-: e 2 - Buie f s

£ * c Constant * 0.5772157

1 IV-*8 G 90 ' n=100' nL

ivv i 2.8660 674 1.9924 818 2.1046 529 11

T

.At Ot-. 1.7755 969 1.9029 058 2.0157 689 12

|

1 6914 486 1.8196 690 1.9332 423 13

r 1.6127 044 1.7418 600 I.856.I 563 14

15 1.5386' 292 1.6687 427 1.7837 772 15

lo 1.4686 252 1.5997 200 1,7155 086 16

17

* 1,4021 995 1.5545 001 I.6508 589 17

18 1.5389 408 1.4720 725 1.5894 186 18

19 1.2785 021 1.4126 915 1.5308 422 19

20 1.2205 878 1.5558 625 I.4?48 362 20

; V 1.1649 442 1.5015 329 1.4211 483 21

/• e ! 1.1115 514 1.2488 842 I.36Q5 608 22

7 1.0596 175 1.1985 258 1.3198 839 2'5

j>4 1-0095 742 1.1494 908 1.2719 514 2h

?5 0.9610 726 1.1022 318 1.2258 163 25

C amputati on Laborst pry

National Bureau of Standard

March 7, 1951