Stati"*5 —1 ^

K«

NATIONAL BUREAU OF STANDARDS REPORT

NUMERICAL COMPUTATION OF LOW MOMENTS OF ORDER STATISTICS

FROM A NORMAL POPULATION

by

J. Barkley Rosser

National Bureau of Standards, Los Angeles, California

and

University of California at Los Angeles

1311

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

1100.-11-5100

NBS REPORT

1311

NUMERICAL COMPUTATION OF LOW MOMENTS OF ORDER STATISTICS

FROM A NORMAL POPULATION

by

J. Barkley Rosser

National Bureau of Standards, Los Angeles, California

and

University of California at Los Angeles

The preparation of this paper was sponsored (in part) by the

Office of Naval Research

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NATIONAL BUREAU OF STANDARDS

Numerical Computation of Low Moments of Order Statistics

1 from a Normal Population

by

J* Barkley Rosser

National Bureau of Standards, Los Angeles, California

and

University of California at Los Angeles

1* Summary* The moments considered in this paper are the first eight

moments, "about the origin, (not including cross moments) of the order

statistics for a sample drawn from a normal population (the precise definition is given at the end of Section 3 below)* These are explicitly expressed

as linear combinations of the moments of the central observations. These,

in turn, are tabulated for samples of twenty or fewer, and an asymptotic

series is provided which will give their values with great accuracy for

samples of more than twenty*

For samples of twenty or fewer, the mean values of the j-th observation

(in magnitude) are tabulated. There is also given an asymptotic formula for

these means which is more accurate than the classic asymptotic formula* For

the means and standard deviations of the extreme observations of a sample,

still more accurate asymptotic formulas are furnished.

2* Algebraic preliminaries. Suppose we have quantities «K(m,n) satisfying the relations

(l) *><(m+l,n) + ©<(m,n+l) - <k (m,n)

4

^The preparation of this paper was sponsored (in part) by the Office of

Naval Research*
.
2

where m and n are any non-negative integers. Clearly a given linear relationship among the f s can be derived from (l) if and only if it would become a polynomial identity upon replacing <?<(m,n) by x

m

(l - x)

n

* We define

Then by induction on N, one can prove for N * 1,

(3)

N

X « x PN

(x(l -x)) - x(l -x) P

N-1

(x(l -x)) •

Multiplying by x

n

(l - x)

n

, we see that from a) can be derived for N * 1

<x(n+N>“) = |

©<(n+l,n) - (N-2) «><(n+2 # n+l )+••+(-!)

(n+l,n+l ) - (N-3) «;(n+2,n+2 )+•••+ (-1

^®<(n+/+l,n+/+l )\ ,

with k as before and

i* N-2

2

Thus, by use of (l) each <*(m,n) with m ^ n can be expressed ^as a sum of

terms of the form ©<(r+l,r) and <?<(r,r).

When there is a symmetry relation

» ^(m,n) = ©<(n,m) '
# i*
3

then one has by (l) that

*(r+l,r) - i «<(r,r) ,

In such case, every s*(m,n) can be expressed as a sura of terms of the

form «<(r,r)#

When there is an antisymmetry relation

©< (m,n) - - (n,m)

then

e*(r,r) * 0

so that every ©<(m,n) can be expressed as a sum of terms of the form

<X(r+l*r) f

3* Choice of basic integrals# Let

2

(4)

(5)

f(x) - (2 e~

x /2

, x

F(x) » J

f (y)dy « -oo

Then

so that

F(-x) - 1 - F(x) ,

^(x) J^(-x)

satisfies (1). So if we multiply by x

s f (x) and integrate* we see that

4

oo

(6 )

o< s (m,n) « Ix

S f(x) F^Cx) F^-xJdx

-00

satisfies (1)« Replacing x by -x in the integrand gives further

( 7 ) oC 2t

(m,n) - *

0+ (n,m) , 2t

( 8 ) 2t+l

Hence every <* 2^(m,n) can be expressed as a sum of terms of the form

<?< 2t(r > r ) every <K 2^+ ^(m# n) can be expressed as a sum of terms of the

form <?<2t+l (r+l,r

)

? Further, we have

±

|

x

2t+1

F

r+1

(x) F

r+1 (-x)j - (2*1) x

2t F^hx) F^h-x)

+ (r+1) x

2t+1 f(x) F^x) Fr+1

(-x)

- (r+1) x

2t+1

f(x) F

r+1

(x) F^-x) .

Integrating from - oo to oo and using (8) gives

( 9 ) °W(r+1 » r ) " /

*

2t pr+1(x) ^+1(-x)dx * -oo

By defining

(10 )

op ^t

(r) - J*

x

2t

F

2* (x) F

2* (-x)dx , -oo

we can rewrite (9) as

^2t+i(r+1 > r ) " r7TMT ft,(r+i)

2 (r+1

)
.
5

Then -we see that every o{ (m,n) can be written as a sum of terms of the form

s *

2t

(r * r) or

(®t (r) * Since the s-th moment, about the origin, of the j-th observation

(in magnitude) from a sample of n is given by

n;

(3 -DS (n-j)l— 8

* (j-l,n-j) ,

we see that evaluation of these moments is reduced to evaluation of the

°< 2t

(r,r) and the ^(r).

4. Numerical determination of the basic integrals* In Table I numerical

values of 4 are given for 1 ^ t * 4 and 1 ^ r * 10. In Table II

numerical values of 4

r ^(r) are given for 0 ^ t ^ 3 and 1 r ^ 10* These

,

\ were mainly computed by numerical quadrature, and it is believed that the

last digit shown is correct to within one unit.

For small values of r, one can get exact values (as in [3]):

*<2t(0*0)

83 1*3 ••• (2t-l)

oc (r.r )

s* ' (2r +l)i

*,(1 , 1 ) - 1 - ,

2 6 2 TvT

(ljl) * —

4 ’ 2 6 7T \/~Z

37

*

2 37T V*

35 820

2 97r \T$

1 1 +

- 3

30 4ir \fl 2 n

2

13 ^2^,2) '

3 « 4ir

6

%(2a 2) - «

8 (2,2) -

(30 (D - <VD *

(32 (D -

?S (D "

f 0

(2) "

(3x

(2) =

?2 (2) -

l» 5

(2) -

74 , . 29 87

S

2 90 40 7T 2 \/T

1640 _ ,, ,> 139 2857

9 2

9 54 150 7T 2 V5

1

Vn

5

evT"

43

20>/F

551

56 V7T

1

2 vTF 7r vTf

arccot \Ts

- (3 (2)

*—

6 0 snV2W — (3 (2) 20 s «

77

eoVrVTn — 6617 ^ (2) - - f 56 9 lOOQTy/nr

We record the values

arccot ]/U * .25268 02551 42075 65548

arccot v"T » <533983 69094 54121 93710 *

To compute ^^.(r^r) and ^.(r) for r >10, we proceed as follows*

Clearly for x ^ 09

o

f(x)(l - 2F(x)

)

7

Integrating from 0 to x gives

1 p

1

e-x

2

(l

w

2

)/2

tF

A -X (1

F(x) F(-x) “r I —

s

dw • J 1 +

Then for 0 4 yt

p( 'FT ) p(-vT)

is completely monotonic (as defined in [1])® Hence

m

4

m

emy/2

|

P(/F) P(_Vy)j

is completely monotonic* So by Theorem B of [1], G-^(y) completely monotonic, where

G

i(y) = h

y

4

m

emy/2 . P(V7) F(-vT)l

m

- e" (4_,r)lIly/27r • .

We infer

0^5, (y) « 0, (0) - ~A^ m

1 1 37r 2

2 2 Multiplying by y exp (-my/2) and putting y « x gives

0Mm

^F(x) F(-x)| - e “2llDC

4 2i2L^J ^4 e-mx

2

/2 # 5?

Putting m ** 1 gives for x ^ 0

8

- 1

V

2 Or -3) _4

2

X

3TT

From this, one quickly derives many of the results of 1*2 3

Since G^(y) is completely monotonic, we -can supply Theorem B of [13

again to infer that Gg(y) is completely monotonic, where

G

2 (y)-^

y

iSLrJl exD /_ (4-^V T . UL -*<>* 120

G-, (y) - m — *-+ ezp

1 3ff d ZT J

~ 30 Tr(Tr- 5

) y

So

But

0 <* G

2 (y) ^ Gg{0) .

g

2 (0)

2 (V-g). m

2 + - 9jr57T -567T

2 + 2807T — 420

105IT

4

(77T

2

-60-7T + 120)

2

2700Tr4

(ir -3)

m

w 0.0000 45737m - 0.0000 12634 m .

Multiplying by y^ exp (-my/2) and putting y

= x

2 gives

2

0 * 4

m

(f(x) F(-x)j

m

- e2mx A

2(7T-3) 4 - ' nil. 7 mx

3 7r mx exp

2uuc

2

7Tr

2 - 607T +120 2 __ x

3or (ir -3)

00005m2 - 0.00001m

j x

8 e"TM*
'
9

Proceeding to make two more uses of Theorem B of [l] gives

s . m

0 6 4

m

j

F(x) F(-x)

j - S(n,x)

6 j^(m) x e 16 -mx

2

/2

where

^(m,x) = e-2mx /tT

. v 4 / 2mx“ v 2 + K^m x exp | — x

+ (K^m2 - K^m) x

8

exp

^ ^

KgX~ - K, m-K,

5 2 x V

4 ' V"

3 + K

9

m2 "

12

m-K,

x exp

2 m-K —2mx __ 2 V

m-K

5 2 2 - K„ — x - —

— x

m “ K

6 (m - K

6

) (K^m - Kgin

2 + K

g

m -K^)

^(m) -

-j K

16

m

10 -K

17

m

y + K

18

m

o .K

lg

m' + K2

0

m

o -K2

1

m

o + K22m".K2

3

m

;

-1 + h/ - K

25

m \W-h?

^- K/* K

9

m -h0\

and — - 0.63661 97723 67581 34308

IT

K. „ HJLzll = 9.5642 23504 51823 8X4 X

10"$ * 37T

2

2 —

_

7 7T -60Tr*12_0 = 4,4537 26195 26472 9694 X 10

2 30ir(TT-3)

-5

K- 4,5737 18562 21895 644 X 10

5

-5

K « 1.2634 27027 83428 618 X 10

4

K,. » 0.71005 42616 12505 98

5

K A„ » 0,27623 62857 80848 29

4 3

-7 - 1,4581 35552 52753 02 X 10

-7

Kc * 1,6111 59796 78083 98 X 10

-»8 K « 7.3706 39743 26849 1 X 10

9

-8 « 1,9600 16255 51239 8 X 10

«9

Kj. - 6.4649 73795 48812 l‘X 10

••8 « 1,4538 54543 52510 69 X 10

-8

1.1327 56568 89158 99 X 10

K * 5,1274 67461 71943 6 X 10~9

Jm<X -1 o

1,5987 53417 33771 X 10

-17

K^

6 « 5.0837 65663 X 10

-16

1.8256 16708 X 10

„ -16 * » 3,1469 17320 X 10

-16

K^b 5.5809 66407 X 10

-16

K^qB 2.9063 68156 X 10

XI

K

gl = 2.6265 40177 X 10”16

f Kgg - 5.6170 52050 X 10

-17

K

25 = 1.1854 05126 X 10“17

K, - 1.5845 19506 X 10

-18

24

-20

K

25 » 6.8859 48529 X 10

We have then

^ .m 0^4 ^(m^x)dx

* tfw 1 • 3 • 5 ••• (2t + 15)

8+t

m

However, we can write

9(m,x)dx

in closed form; indeed the leading (and dominant) term is

1 5 » 5 »» (2t - i ) *n't*^

(4m \/ 2m

Also for m ^ 11,

fl(m) 4 (4.2) m

4 10

-10

- .

Thus for m ^ 11, we have quite an accurate approximation to 4

m ^(m)*
.
12

If a more accurate approximation is desired, one can carry out a numerical quadrature of the function

x

2t

^4

m

(f(x) F(-x)} - e(m,x)^

For large m, this function is both small and very smooth, and one can

readily get a high accuracy by a simple numerical quadrature#

In a similar fashion, we can get accurate numerical approximations

for the ^2t

(r,r)«

5* Estimates for means. In computing moments of arbitrary observations from the moments of central observations by the formulas given in

Section 2, it is clear that there will be considerable loss of significant

figures for the outlying observations when the sample size is large. We

now have sufficiently accurate values for the central observations that

this effect is not damaging for samples of reasonable size. Thus, we used

this method to compute the means of observations for samples of size twenty

or fewer. The results are tabulated in Table III, where m(j|n) denotes the

mean of the j-th observation from a sample of size n. We have retained as many decimals as we think are accurate, but do not guarantee the final

digit shewn. Our table extends .the tables of means given in [3] and [4].

Clearly, one could use our estimates of ^Q

(r ) to extend the accuracy of

Table XX of [5].

Incidentally, in making up such a table, one would start with the

«<(r,r) and o<(r+l*r) and work outward by means of the relation (1). Our

explicit formula for ©<(n+N,n) would be used only to estimate the accumulation of errors, or to compute an isolated value.
. ’

-
13

Quite clearly, the method outlined above for computing moments is

essentially a recursive procedure, in that the computation of means for a sample of a hundred would require the computation of a great many means

for observations from smaller samples* Also for large sample sizes, the

loss of significant figures begins to be serious for the outlying observations* For these reasons, it seems worthwhile to provide some way of

estimating means of observations in a large sample, particularly the outlying observations#

For m(jjn), there is a classical asymptotic estimate, which we shall

denote by ^(j|n). It is defined by the equation

Some discussion of this estimate is given in [4] and [6]. Values of

sample of size twenty, the estimate is rather poor, especially for outlying

observations

To obtain a better estimate, we start by inquiring for the value of

8 which minimizes

y(j|n) are tabulated in Table III, from which it is clear that even for a

(11 )

n

00

/ (x- £ )

2

f(x) F^

-1

(x) F^O-x) dx • -00

Clearly (11) can be expanded to

- 2 S’e<

1 (j-l,n-j) + $

2 (j-l)i (n-j)t

«
.
14

So (11) is a minimum -when

nJ

(j-l)I (n-j)I — ^i^-l^n-j) * m(j|n) .

That is, m(j|n) is the value of o which minimizes (11)* So we estimate

m(j|n) by seeking a value of $ which makes (11) small* To this end, we

first observe that

(12) f(x) F^

-1

(x) F

n“',’(-x)

is non-negative, with a single maximum. If (12) were exactly symmetrical,

then the x which maximizes (12) would obviously coincide with the $ which

minimizes (11)* Actually, (12) is reasonably symmetric about its maximum

for many values of n and j. Thus one is tempted to use the x at which (12)

takes its maximum as an estimate for the S at which (11 ) takes its minimum.

Accordingly, we define S(j|n) to be the value of x at which (12) takes its

maximum, and hope that thereby £(j|n) will be a good estimate for m(j|n).

We have listed values of $(j|n) in Table III. For all entries in

Table III, we have

Jr(d|n) 6 S(j|n) ^ m(j|a) .

One is led to conjecture that this holds for all j and n, so that £(j|n)

is a better estimate for m(j|n) than ^f(j|n).

From Table III, one is tempted to make still another conjecture, namely

that except for outlying observations, S ( j

| n

)

lies about halfway between

y(j|n) and m(j|n). Thus we are led to suggest

2 S(j|m) - jr(j|n)
.

I

.

.
15

as an estimate for m(j|n) f This estimate is also tabulated in Table III.

The four decimal values listed for 2 £ » may be off by as much as three

units in the fourth decimal place, since they were computed directly from

the four decimal values of ^(j|n) and S(j|n).

From Table III, it appears that 2 £ - ^

is a fairly successful estimate

for m(jjn) except for the outlying observations, for which it does not

furnish a good estimate. The reason for this is fairly clear. As n oo , the quantity S(n|n) - ^(njn) approaches zero fairly rapidly. Thus If, $ * and 2 £ - are all about the same for large n. However (see [?]), the

asymptotic formula for m(njn) is

(13) ^ n f ( %

(n | n ) ) *

where C is the Euler constant. For large n, y(n|n), £(n|n), and

n f ( ^

(njn)) are all nearly equal, and all go very slowly indeed to oo as n goes to oo • Thus £, y, and 2 J - jr all become nearly equal before any

of them is a good approximation to m(njn).

For outlying observations near the extreme, the situation is similar,

but becomes less severe as one moves further from the extreme (see equation (6) of [7])#

For the extreme observation, an approximation was proposed by Bortkiewicz

(see [8]), namely

B(n) = i

Sample values of this are listed in Table IV, where for comparison are also

listed values of m(n|n). For n > 20, the values of m(njn) are taken from

Tippett’s table (see [9]).

^£(n|n) + f ( ^

((n-l)| (n-1))) \/n(n-l)

j

16

We wish to propose an alternative formula for m(n|n). Our formula is

better than Bortkiewicz * s formula for smallish n, say n ^ 100, but Bortkiewicz*s formula is better for large n, at least as far as n » 1000*

One advantage of our formula is that it is less empirical than Bortkiewicz*s. Indeed, one may use a similar derivation to get similar estimates

for other outlying observations# Furthermore, although our formula appears

eventually to become inferior to Bortkiewicz *s, the discrepancies are far

more uniform, and can be explained. Then by curve fitting methods, one can

get a formula which gives a really close fit for 2 ^ n ^ 1000, and which can

be expected to continue to give a close fit for larger n#

We recall that m(n|n) is the value of F which maximizes

r

00

(x - £ ) 2

f(x) F

n

”^(x) dx.

-oo

Write temporarily $ = F (n|n), so that

(14) ^ F(jZf) = (n -1) f (^) .

Then m(n|n) - $ is the value of Q which minimizes

r

00

I ((x-^)«3)2

f(x) F

n~M

(x) dx • -00

But this value of Q is given by

f > ( \

r

00

r°

Q a -

I (x-jzf) f (x ) F

n"1 (x)dx •j / f(x) F^COdx

u-CO

J

1 -00

V.

17

To estimate this, we expand the integrands as power series in x - ^

Using (14), the first few terms are given by

f(x ) (i+iz) t (

a - 1 12

\ “-1/

6 \

(n-l)2

i +*

Thus we are led to propose the approximation

exp /-&=£! UM\ " n-l f(x)

n-l

J

\x+te£L 1

1

- 1

6 \ (n-l)

2

Using this approximation in both the integrands of our formula for 0,

we get the approximation

« = l(/r.(n+l)- (n-l)

2

) .

' 2 (/^

2

n + n-l )

2

Thus we are led to the approximation for m(n|n) given by

C(n) = J (n|n) - 1 + (f (n[n))2n(nvl)- (n-l)'

2(( J (n|n))2

n + n-l)2

Values of C(n) are listed in Table IV#

For very large n, we have approximately

0(" ) ' s<° w *rFiM

Comparison with (13) shows that for large njC(n) will be less than m(n|n),

since ^(n|n), f (n|n), and n f( jj*(n|n)) are all nearly equal for large n#

Indeed, from Table IV, one would conjecture that C(n) < m.(n|n) for n & 7*

In view of (13), one sees that for very large n a better formula than

C (n ) woul d be

S (n | n ) + 2C fc(n)- $(n|n)j

18

Fitting empirically the difference between this and m(n,n), we are led

finally to propose

D(n) = J (n jn) 1 + C

( r (n|n))2

n(n+l) - (n-1)

2

(( f (n|n))2

n + n-l)2

19t541 ( $ (n|n)) - 66.366 S(n|n) + 114.448

Values of D(n) are listed in Table IV.

The numerical coefficients in the denominator of the second term of

D(n) were chosen to give a fit in the region 2 ^ n ^ 100. Clearly, they

could be chosen to give a fit in any other region. However, in view of

(13), one might expect high accuracy for D(n) for all values of n. If one has a good estimate for the mean of the extreme observation,

one can derive a good estimate for its standard deviation by using the

empirical formula proposed in [10]«

Incidentally, by the time one gets to n = 1000, the asymptotic formula

(13) agrees with m(n|n) to within less than one per cent. One would suspect

a similar accuracy for (6) of [7] for other small values of m.

, 6. Acknowledgements. The computations for the numerical values given

in this paper were supplied by the National Bureau of Standards 9 Institute

for Numerical Analysis. The numerical quadratures were performed on IBM

equipment by Albert II. Rosenthal, Mrs. Merry Kruse and Frederick Hollander

under the direction of Dr. Everett Yowell. The values of y(j|n) and

S(jl n ) were computed by Mrs. Louise W, Strauss, the were computed by

Mrs. Nan Reynolds and checked by Miss Gladys P. Franklin, and the B(n) and

C(n) were computed by Mrs. Nan Reynolds. Dr. Arnold D. Hestenes helped with

the checking and exercised the overall supervision for much of the computation.

October 17, 1951
.

.
TABLE I

Values of 4r°Wr’ r >

r t 1 t « 2

1 0.2991 1406 9718 8053 0*4072 7207 9892 6007

2 0.1629 7795 2856 4674 0.1330 4219 4614 9201

3 0*0962 0427 9558 9977 0.0612 8897 0422 4789

4 0*0674 9512 3853 7019 0.0338 9389 5809 0415

5 0.0506 6895 0920 9147 0.0209 8985 6215 1131

6 0.0398 2756 2731 0725 * 0.0140 3808 4528 5253

7 0.0323 6529 0981 3039 0.0099 2626 8505 3476

8 0.0269 7240 6234 8521 0.0073 2100 7671 5223

9 0.0229 2631 2941 7515 0*0055 8079 0652 7179

10 0.0197 9919 4831 1596 0.0043 6874 1679 6848

"fc a 3 t » 4

1 0.9337 0260 8619 4196 3.0237 4900 0611 929

2 0.1947 6555 2031 6235 0.4028 7649 7414 444

3 0.0656 4164 9173 2218 0.0992 4109 4940 865

4 0.0285 8077 9942 2641 0.0339 8616 3643 807

5 0.0145 8715 6381 2597 0.0142 8346 2227 132

6 0.0082 9464 7482 9277 0.0069 0054 1881 627

7 0.0051 0028 1317 1660 0. 0036 8762 1539 726

8 0.0033 2741 7523 9426 0* 0021 2706 2546 631

9 0.0022 7385 5368 6564 0.0013 0254 5317 193

10 0.0016 1294 4832 8192 0.0008 3694 8262 323 '
*

'
TABLE II

Values of 4

r ^.(r)«

r •fc a 0 t - 3

1 2.2567 5833 4191 0251 1.8806 3194 5159 1876

2 1.5840 6070 5464 7751 0.6427 8513 7922 0755

3 1*2898 9973 6330 9072 0.5455 8405 5478 4261

4 '1.1155 3389 5536 3525 0.2228 1607 0880 1860

5 0*9969 1887 5707 6363 0® 15 8 7 7144 7923 6560

6 0.9095 3967 2162 7550 0.1204 4214 6659 3750

7 0.8417 2526 9829 5399 0. 0955 8452 3113 145 9

8 0.7871 1902 6160 2866 0.0779 5172 7041 1192

9 0.7419 2432 8542 4815 0.0652 4962 7985 1744

10 0.7037 1536 6288 9335 0.0556 5770 1650 7985

"fc 53 2 t « 3

1 4.8520 3041 8510 7041 21.5989 0491 8847 042

2 0.7982 5853 4383 1039 1.6825 9501 5448 810

5 0.2814 2174 8929 4527 0.3873 8273 5294 494

4 0.1349 9461 2115 8106 0.1377 7182 9511 448

5 0.0765 4181 1168 1755 0.0620 3995 4413 791

6 0.0482 0967 3030 9395 0.0324 0067 7160 751

7 0.0326 3892 7355 2676 0.0187 3590 2424 600

8 0.0232 9272 7472 9138 0.0116 6593 9436 320

9 0.0175 0364 7795 6415 0.0076 8672 8404 898

10 0.0132 6717 8669 9766 0.0052 9499 0746 468

TABLE III

Table of accurate and approximate means

o

n 3 m(j|n) yli |n) S(j|n) 2S-y

2 2 0*5641 89583 54775 62869 0.4307 0.5061 0.5815

3 3 0.8462 84375 32163 44304 0.6745 0.7653 0.8561

4 4 1.0293 75373 00396 41321 0.8416 0.9359 1.0302

4 3 0.2970 11382 27464 53255 0.2533 0.2775 0.3017

5 5 1.1629 64473 64051 96128 0.9674 1.0615 1.1556

5 4 0.4950 18970 45774 22092 0.4307 0.4654 0.5001

6 6 1.2672 06360 61147 12976 1.0676 1.1602 1.2528

6 5 0.6417 55038 78576 11884 0.5659 0.6058 0.6457

6 4 0.2015 46833 80170 42508 0.1800 0.1919 0.2038

7 7 1.3521 78375 60690 43992 1.1504 1.2412 1.3320

7 6 0.7573 74270 63887 26882 0.6745 0.7171 0.7597

7 5 0.3527 06959 15298 24389 0.3186 0.3371 0.3556

8 8 1.4236 00306 04527 775 31 1.2207 1.3095 1.3983

8 7 0.8522 24862 53829 09223 0.7647 0.8087 0.8527

8 6 0.4728 22494 94061 79858 0.4307 0.4533 0.4759

8 5 0.1525 14399 50692 31941 0.1397 0.1467 0.1537

9 9 1.4850 13162 20923 70063 1.2816 1.3684 1.4552

9 8 0.9322 97456 ‘73360 37275 0.8416 0.8863 0.9310

9 7 0.5719 70782 85469 61041 0.5244 0.5495 0.5746

9 6 0.2745 25919 11246 17493 0.2533 0.2649 0.2765

10 10 1.5387 52730 83517 28560 1.3352 1.4202 1.5052

10 9 1.0013 57044 57581 43585 0.9085 0.9534 0.9983

10 8 0.6560 59105 36476 12033 0.6046 0.6313 0.6580

10 7 0.3757 64696 99787 75394 0.3488 0.3633 0.3778

10 6 0.1226 67752 28435 80642 0.1142 0.1188 0.1234

11 11 1.5864 36351 90800 01689 1.3830 1.4662 1.5494

11 10, 1.0619 16520 10689 97268 0.9674 1.0124 1.0574

11 9 0.7288 59404 68593 02011 0.6745 0.7023 0.7301

IX 8 0.4619 78307 17497 72089 0.4307 0.4474 0.4641

11 7 0.2248 90879 18795 31177 0.2104 0.2183 0.2262

12 12 1.6292 27639 87191 29903 1.4261 1.5076 1.5891

12 11 1.1157 32184 30495 91343 1.0201 1.0648 1.1095

12 10 0.7928 38199 11660 26895 0.7363 0.7649 0.7935

12 9 0.5368 43021 39391 27361 0.5024 0.5206 0.5388

12 8 0.3122 48878 75710 61544 0.2934 0.3036 0.3138

12 7 0.1025 89679 81913 88664 0.0966 0.0999 0.1032

13 13 1.6679 90177 04912 74980 1.4652 1.54.52 1.6252

TABLE III (continued)

Table of accurate and approximate means.

n 3 m( j|n) ycjja) <SX j|n) 2S- y

13 12 l e 1640 7719 3 74 53 3 889 72 1*0676 1*1120 1*1564

13 11 0*8498 34632 38287 04386 0*7916 0.8207 0*8498

13 10 0o6028 50088 22904 35257 0*5699 0*5852 0.6045

13 9 0.3883 27121 01486 84596 0*3661 0*3780 0*3899

13 8 0*1905 23691 09268 64661 0*1800 0*1857 0*1914

14 14 lo7033 81554 09997 65215 1*5011 1*5796 lo658l

14 13 1*2079 022 7 5 38809 019 31 1*1108 1*1548 1.1988

14 12 0.9011 26703 88883 11217 0*8416 0*8709 0*9002

14 11 Do 66 17 63703 52768 12673 0*6229 0*6430 0.6631

14 10 0o4555 66049 98244 91716 0*4307 0*4440 0*4573

14 9 0o2672 97048 87322 31781 0.2533 0*2609 0*2685

14 8 0*0881 59214 05197 08501 0.0837 0*0862 0*0887

15 15 1*7359 13444 94103 74337 lo 5341 1*6113 1.6886

15 14 1 ® 24 79 3 508 2 3 2 512 37 511 1*1503 1*1940 1*2377

15 13 0*9476 89030 29737 20662 0*8871 0*9166 0*9461

15 12 0*7148 77398 25466 73439 0*6745 0*69 52 0.7159

15 11 0 0 51 57 01043 02846 9 5569 0*4888 0.5030 0*5172

15 10 0*3352 96063 89040 84009 0*3186 0*3278 0.3370

15 9 0 0 16 52 98526 34744 53440 0*1573 0.1617 0*1661

16 16 1*7659 91393 05478 79673 1*5647 1*6406 1.7165

16 15 1*2847 44223 23477 94295 1*1868 1*2301 1*2734

16 14 0*9902 7109 5 9 5753 40023 0*9289 0*9 584 0*9879

16 13 0o763l 66745 77000 36761 0*7215 0*7426 0.7637

16 12 0*5700 09355 70865 83470 0*5414 0*5564 0.5714

16 11 0*3962 22755 13205 42186 0*3774 0*3875 0 * 39 76

16 10 0*2337 51578 48766 53715 0*2230 0.2289 0.2348

16 9 0.0772 87459 31001 9 5943 0*0738 0*0757 0.0776

S 17 17 lo7939 41980 88269 08735 1*5932 1*6679 1.7426

i i? 16 l e 3l87 81987 80834 14671 1*2206 1*2635 1*3064

17 15 1*029 4 609 8 8 9 3306 41474 0*9674 0*9969 1*0264

1.7 14 0*8073 84928 73839 33255 0*7647 0*7861 0.8075

17 13 0*6194 57651 12273 73158 0*589 5 0.6051 0*6207

17 ' 12 0*4513 3344S 71486 88219 0*4307 0.4417 0*4527

17 11 0*29 51 8648 7 23022 74 458 0*2822 0*2892 0.2962

17 10 0 « 14 59 87423 14114 81225 0.1397 0*1431 0*1465

n

18

18

18

18

18

18

18

18

18

19

19

19

19

19

19

19

19

19

20

20

20

20

20

20

20

20

20

20

TABLE III (conoluded)

Table of accurate and approximate means

.

3 “(i! n) *(3|n) 2 <5 ~ Y'

18 1.8200 31878 96872 21046 1.6199 1.6934 io7669

17 1 o3 504 13713 42015 99452 1.2521 1.2946 1.3371

16 1,0657 28182 91379 36422 1.0021 1.0326 1.0631

15 0,8481 25019 02941 66731 0.8046 0.8262 0,8478

14 0o6647 94612 71981 16086 0.6336 0.6497 0,6658

13 0,5015 81550 97034 41543 0.479 5 0.49 12 0.5029

12 0,3508 37238 203S1 81572 0.3360 0.3440 0.3520

11 0.2077 3 53 07 12871 34707 0.1992 0.2038 0,2084

10 0.0688 02568 15669 14373 0.0660 0,0675 0.0690

19 1.8444 81511 60382 46581 1.6449 1.7173 1.7897

18 1.3799 38491 53687 61420 1.2816 1 o3236 1.3656

17 1.0994 53099 42807 22722 1.0364 1.0658 1.09 52

16 0,8858 61961 50430 76155 0.8416 0.8634 0.8852

15 0,7066 11484 74857 5639 3 0.6745 0.6909 0o7073

14 0.5477 07371 03927 23228 0.5244 0.5367 0.5490

13 0.4016 42274 15433 31226 0.3853 0.3941 0.4029

12 0.2637 42890 86034 96451 0.2533 0 • 2 59 0 0.2647

11 0.1307 24879 49771 37309 0.1257 0.1284 0,1311

20 1.8674 75009 79832 04847 1.6684 1.7398 1.8112

19 1.4076 0409 5 90840 39 521 1.3*092 lo3508 1 o 39 24

18 1.1309 48052 19312 58 507 1.0676 1.09 68 1.1260

17 0.9209 81700 42610 19940 0.8761 0,8980 0 o9l99

16 0.7453 83005 81713 01013 0.7124 0.7292 0.7460

15 0.0908 96921 54291 22532 0.56S9 0.5787 0. 5915

14 0.4483 31753 19744 58187 0.4307 0.4401 0.4495

13 0.3149 33241 64569 52584 0.3030 0.3094 0.3158

12 0.1869 57364 68233 12250 0.1800 0.1838 0.1876

11 0.0619 96286 49429 2349 3 0.0597 0 ,0610 0.0623
'
TABLE IV

Table of accurate and approximate extreme means.

n m(n|n) B(n) C(n) D(n)

2 O.pObB 0.5351 0.3634 0.3629

3 0.8 1|-6 3 0.8280 0.8474 O.8I467

4 1*029 U 1.0183 1.0301 1.0302

5 1.1630 1.1568 1.1633 1.1639

8 1.2672 1.2644 1.2672 1.2681

7 1.3522 1.3318 1.3519 1.3529

8 1.1(236 1.1-1250 1,4231 1.1(21(2

S l.483o 1.4870 i ,4s 42 1.4833

10 1.5388 1.51(26 1.5377 1.5391

11 1.586U 1,5911 1.3832 1.5067

12 1.6292 1.6345 1.6278 1,6294

13 1.6680 1.6738 1.666 ll. 1,6681

lU 1.7034 1.7097 1.7017 1.7034

i5 1.7359 ' 1. 71)26 1.7340 1.733 s 16 1.7660 1.7729 1.7640 1.7659

17 1.7939 1.8011 1.7918 1.7937

18 1.8200 1,0274 1.8170 1.8198

19 1 .8443 1.8520 1.81(21 1.841(2

1 20 1.0675 1.8752 1.8650 1.8671

30 2.0l|28 2.0309 2.0395 2.0422

4o. 2.1608 2.1687 2.1570 2.1601

3

0

2 e 2 li9

1

2.2567 2.2 U49 2. 2484

70 2.3774 2.3842 2,3726 2.3767

100 2.5076 2.5134 2.3023 2.5071

200 2.7460 2.7496 2.7398 2.7458

300 2.8778 2.8800 2.8711 2.8777

3 oo 3. 0367 3.0374 3.0296 3.0368

700 3.1376 3.1374 3.1301 3.1378

1000 3.2 U1 U 3.2403 3.2338 3.2418

References

[1] J. Barkley Rosser, "The complete monotonicity of certain functions

derived from completely monotone functions,” Duke Math. Jour.,

Vol. 15 (1948), pp. 313-331.

[2] J* D. Williams, "An approximation to the probability integral,”

Annals of Math. Stat., Vol. 17 (1946), pp. 363-365.

[•3] H. J. Godwin, "Some low moments of order statistics,” Annals of Math.

Stat., Vol. 20 (1949), pp. 279-285.

[4] C. Hastings, Jr., F. Mosteller, J. W. Tukey and C. P. Winsor, "Lowmoments for small samples; a comparative study of order statistics

Annals of Math. Stat., Vol. 18 (1947), pp. 413-426.

[5] R. A. Fisher and F. Yates, Statistical Tables , Third Edition, Oliver and

Boyd, London, 1948.

[6] Frederick Mosteller, "On some useful inefficient 1 statistics," Annals

of Math. Stat., Vol. 17 (1946), pp. 377-408.

[7] E. J. Gumbel, "Ranges and midranges," Annals of Math. Stat., Vol. 15

(1944), pp. 414-422.

[8] L. von Bortkiewicz, "Variationsbreite und mittlerer Fehler,"

Sitzungsberichte d. Berliner Math. Gesellschaft, Vol. 21

(1921), pp. 3-11.

[9] L«. H. C. Tippett, "On the extreme individuals and the range of samples

taken from a normal population," Biometrika, Vol. 17 (1925),

pp. 364-387.

CLO] H. E. Daniels, "A property of the distribution of extremes," Biometrika,

Vol. 32 (1941), pp. 194-195
:: ' • * • 
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