,
"';;,.~$'~,I
MOMENTS OF ORDER STATISTICS FROM A NORMAL POPULATION
by
•
R. C. Bose and Shanti S. Gupta
Univeraity of North Carolina,
.~
".,
_.
..
This research was supported by the 'United States Air Force
through the Air Force Office of Scientific Research of the
Air Research and Development Command under contract No.
AF
18
(600) -
83.
Institute of Statistics
Mimeograph Series No. 154
July, 1956
-till
'"t".o
UN CLASSIF18D
Security Information
Bibliographical Control Shoet
1. University of North Cdrolino, Ctepel Hill, N. C.
2. M'thom!.ltics D'v1s1on, Air Forco Office of SciElntific Reseerch
3. AFOSR-TN-S6-501
4. ASTIA AD 110 316
5. MOHTIrl:S
QIi' ORD08R STJlTIST:;:CS li'ROH A NORIvLU.r, POPULATION
6. R. C. Bose and Shanti S. Gupta
7. October, 1956
8.
!\F
18(600)-83
9. FHa 3.3
MOMENTS OF ORDER STATISTICS FROM A NORMAL POPULATION
By
R. C. Bose and Shanti S. Gupta
University of North Carolina
1.
Introduction and summary.
Statistics based on ordered observations have been called systematic statis-
tics by Mosteller ~12_7. They are now being increasingly used in new statistical
procedures ~l, 2, 3, 4, 7, 8, 13, 14, 15, 16, 17, 20, 22_7.
The present paper
deals with the problem of obtaining the moments of X(k)' the k-th order statistic
for a sample of size n from a normal population N(O, 1).
This problem has been
considered among others by Cole ~5_7, Godwin ~6_7, Hastings, Mosteller, Tukey
•
and Winsor ~9_7, Jones ~1l_7, Ruben ~18_7 and Tippett ~23_7.
~t(n,
It has been shown that
k), the t-th moment of X(k)' can be expressed
in terms of lower moments of order
t - 2i
(i = 1, 2, ••• , t/2 or (t - 1)/2) and
the integral
+00
J
(1.1)
2
-(t+l)x /2 d
Pt+l ()
x e
x
-00
where Pt+l(x) for t
> 0,
is defined by
:
,
(1.2)
it being understood that in (1.2),
the c. d. f. of N(O, 1).
••
n
<t
and is zero if n
when n
n
,41'
< 6.
~
~
is replaced after differentiation by
Pt(x) is thus a polynomial of degree (n - t) in
> t.
~(x),
~(x)
if
Exact values of all odd order moments can be derived
5, and the exact values of all even order moments can be derived when
Godwin ~6_7 and Jones ~11_7 have given tables of exact moments
~t(n, k) for t = 1 and 2. The corresponding tables for t = 3 and 4 are provided
2
4It
in this paper.
In general the numerical evaluation of the integral (l.l) can be
expeditiously done by using the Gauss-Jacobi method of mechanical quadrature
~21_7 based on the zeros and the weight factors of the Hermite-polynomials for
which tables have been provided by Salzer, Zucker and Capuano ~19_7.
It is be-
lieved that the formulae derived here are better suited for numerical computation
than those given elsewhere.
2.
The function Pt{n, k, x).
Let xl' x 2 ' ••• , x be n independent observations from a normal population
n
N{O, 1) with zero mean and unit variance, and let
(2.1)
•
be the n ranked observations among xl' x '
2
..., Xn •
Then the cumulative distribu-
tion function of X{k)' the random variable corresponding to x{k)' (I ~ k ~ n), is
given by
(2,2)
J
X
b(xlf-1f1 -
~(x)]-k
-00
where w{x) is defined as
S
X
\
¢(X)
e _x
2
/2
-00
..
and C is the constant
(2.4)
.e
C
n~
= -r.---,;-"I:-:--.,....-~
(k - I) ~ (n - k) ...~
dx
2
e- x /2 dx
UN CLASSIFIED
Security Information
Bib1iog~aphica1
Control Shoet
1. University of North C8ro1ino, Ctape1 Hill, N. C.
2. M-tl1omotics D'v1s1on, Air Force Offico of Scientific Research
3, AFOSR-TN-S6-501
4.
ASTrA AD 110
'). MOMENTS Oli'
316
ORD~
STATISTICS FROM A NORMAl. POPULATION
6. R. C. Bose and Shanti S. Gupta
7. October, 1956
8. AF 18(600)-83
9. F:ile 3.3
3
Let us now define the function pt(n, k, x), which we shall abbreviate to
Pt(x) for convenience, by the relation
where PO(x) is given by (2.2).
Then
(2.6)
It is clear that Pt(x) is a polynomial of degree n-t in ¢(x) if 1
zero for t
> n.
t
S n,
and is
In fact, we can write
P
()
t+l x
•
~
=C
t
~ rl-l(l _ ¢ )n-k 7
d¢t 1..
-
where ¢ is replaced by ¢(x) after the differentiation.
t, n, k, ¢t(x) is a bounded function of x.
The functions P2 (x), P (x), P4 (x) and
3
P (x) are given below explicitly, where ¢ is written for
5
= C cI> k-2 (1
8
(2.)
P2 (x )
(2.10)
P4 (x) = Ceil k-4(1 -
-
¢
It follows that for given
)n-k-lr
1.. (k-l) - (n-l)
~(x).
7
~
~)n-k-3L(k-l)(k-2)(k-3) - 3(k-l)(k-2)(n-3)¢
+ 3(k-l)(n-2)(n-3)~2 - (n-l)(n-2)(n-3)<1»_7
,
(2.11)
P (x)
5
= C ¢k- 5 (1
-
~)n-k-4L-(k_l)(k_2)(k_3)(k_4) - 4(k-l)(k-2)(k-3)(k-4)1l
n
+ 6(k-l)(k-2)(n-3)(n-4)¢C:
••
- 4(k-l)(n-2)(n-3)(n-4)<1>3 +
3.
.e
,
A system of differential equations satisfied by PO(x) •
From (2.5)
(n-l)(n-2)(n-3)(n-4)¢~
4
(3.2)
In general let us assume
Pt ()
x
=
dt-rp
2
( c:.1t )t/2 e tx /2
°
0
dyt-r
where
(3.4)
•
and g
Differentiating (3.3) and
t(X) is a polynomial in x of the r-th degree.
r,
using (2.5), we have
=
(21t) ( t+ l) /2 e ( t+ 1 h/~ /2 til
1'=0
~
g
dt-r+l p
t(x) . '.
~+
~'dxt-r+
+
t
txg
r,
t(x)
~,t(x) }
This leads to the recurrence relation
(3.6)
..
•.
where g
1',1'
(x) should be interpreted as zero.
all the polynomials g
r,
t(X).
"
~
,WI'
\
Starting from
gO,l (x)
we can successively calculate
=1
This together with (3.4) determines
5
(3.8)
go , 2(x)
= 1,
(3.9)
gO,3(x)
= 1, gl,3(x) = 3x, g2,3(X) = 2x 2+1
gl , 2(x) . = x
= 1,
gl , 4(x)
(3 .11) go, 5 (x) = 1,
gl,5(X)
(3.10) go , 4(x)
= 6x ,
g2 , 4(X)
= llx2+4, g3,4(X) = 6x\7x
2
= lOX, g2,5(X) = 35x +10, g3,5(X) = 5OX\45X
g4,5(x)
= 24x4+46x 2+7 .
Hence we have the set of equations
1
(2rc)
1/2 e
-x
2
/2
•
P l (x)
1
(2rc )
2
372 e
-3x /2
P3 (x)
d2 p
dP
2
030
(llx +4)~ + (6x +7X)dx =
dx
We can proceed in this manner up to any order but it should be noted that
Pn(x) is a constant and Pt(x)
=a
if t
> n.
The general equation is
·.
•
••
.e
4.
Moments of X(k) •
We shall first prove the following Lemma:
6
If a and r are non-negative integers, then
Lemma.
(4.1)
or 0
according as
r<o
or
r>a
where ~~-r is the (a-r)-th order moment of X(k) about the origin.
It should be noted that by definition
(4.2)
•
I
~a
•
/2
)
PI (x + -2rc e
From (3.12) and (3.13)
(4.3)
2
d p
(4.4)
o
-2
dx
=-
x
(2rc)
1/2 e
-x
2
1
-x
2
and in general using the system of equations (3.12)-(3.17) we can write
r
E
2
i=l
where h. (x) is a polynomial in x.
1
.
-,
~
.
.e
+'00
(4.6)
fp
-00
1
1
Now
+00
dr+l p
0
dx
dxr + 1
=
l-
h. (x)e -ix /2 p . (x)
a drpo
x -
dx
r
1
- a
+00
J
r
a-I d Po
ax
-d
r x.
dx
-00
Since Pt(x) for any non-negative integer t is a bounded function of x, it
follows from (4.5) that the first part on the right hand side of (4.6) vanishes.
7
<0
Repeating this process we get if r
+00
+00
f
-00
If r
f
r
(-1) 0(0;-1) ... (o;-r+l)
dP
0-( -1 )r0;(0;-1 ) ••• (o;-r+ 1 ) !le'
xo;-r -xd
t-r •
-00
> 0,
we get on repeating the process 0; times,
+00
f
dxr + 1 -a
-00
dx
-00
+00
=
(-1)00;(0:-1) .•• 3.2.1
-00
=0
•
This
pr~~es
.
the Lemma.
On applying the Lemma and integrating the equations (3.13) ••• (3.16) we get
+00
III =
(4.7)
1
21t
f
P2(x)e-
x
2
dx
-00
+00
-3 + (112 + 1)
(4.8)
=
f
1
(21t) 3 72
')
P~ (x)e _3xc. /2 dx
J
-00
+00
_2'.::\·
I
~1
+
(4r 3I
+
1
lL1I ) = ---,.,.
( .~<:::n: )2
f
-00
+00
(4.10)
--.
~
.e
70 - 15CJl2 - 45 + (24~ + 46~ + 7)
f
-00
We may write
~(n,k)
instead of w~ to denote the fact that we have thea -th moment
about the origin of the k-th order statistic out of a sample of n observations
from N( 0,1).
We then have
8
+00
(4.11)
I-li(n,k)
1
= 21!
f
dx
-00
'(
1-12
n,
k)=l+
1
2t(2n)3/2
+00
(4.15)
1-l '(n,k)
3
~ 2
2 1-I1'(n,k) +
1
3 Q2rc)2
f
P4(x)e- 4x
2
/2
dx
-00
f
+00
(4.14)
4 + "3
13 1-l (n,k) + ---::5"...·/'
11:!!:"1J
= - '3
2
4 ~ (21C) c;.
2
P5(x)e -5x /2
dx
-00
In general applying the Lemma to (3.17) we can express ut(n,k) in terms of
4IIIt
lower moments of even (odd) order when t is even (odd) and the integral
+00
f
(4.15)
-(t+l)x
Pt+l ()
x e
2
/2
d
x
-00
where the polynomials P (x) ••• P (x) are given by (2.8) ••• (2.11).
2
5
, xn ,
cular case when n = k, i.e" x(n) is the largest of xl' x2 '
In the parti-
...
the very simple form
(4.16)
Pt (x)
= n(n-l) (n-2) '" (n-t+l) [cfJ(x)}n-t ,
Hence we get
(4.17)
.
.-,
.e
2(~)
I-li(n,n) = 2rc
+00
f
[)(x)r-
2
e-
x
2
dx
-00
(4.18)
1-1 (n,n) = 1 +
2
n
3(3)
(2rc )3/2
+00
J
-00
1-cfJ (x )1n - 3 e -3x 2/2
-
-
dx
9
+00
(4.19)
~ l=e(xlI n-4 .-4x
4(4)
~3'(n,n) ~ ~2~1'(n,n)
+
(2n)2
2
J2
dx
-00
f
+00
(4.20)
~4(n,n)
=
5(n)
4
13 , ( )
5
- -3 + -3 lS2- n,n +
"'/2
(2n»)
. 1-_"'(x)In-5 e-5x
'¥
.
2
/2
dx
-00
It should be noted that in the formulae (4.17) through (4.20)
should be interpreted as zero if t
> n.
Some integrals of the type occurring in (4.17) through (4.20) have been
numerically evaluated by Hojo
•
/_-1o_7.
5. Exact values of some moments •
Let
+00
I (a)
n
(5.1)
f
=
-00
then
I
Now
•
J
(a)
°
=
1'(
1/2
+00
2 1
[e(ax) _~ 1 m+-
2
e -x
dx
= 0,
-00
since the integrand is an odd function of x.
•.
2m+l
1 2rn+ 1 (a)
~
.e
In particular
=
E
r=l
(_l)r+l
Hence
10
and
In general, I 2mt1 (a) can be expressed as a linear function of 1 2m (a),
1 2m _2 (aL ... , IO(a).
Differentiating (5.1) with respect to a, (this is justified in virtue of the
uniform convergence of the integrals with respect to a,
-00
< a < 00,
and the conti-
= 2,
nuity of the integrands), we get for n
(5.7)
60
•
that
(5.8)
Using (5.6)
(5.9)
Since Pt+l(x) is a polynomial in
~(x)
of degree n-t-1, by using (5.2), (5.5),
(5.8) and (5.9), we can exactly evaluate (4.15) if n
evaluate
n ~ 6.
=1
t
n k
•
••
.e
~t(n,k)
for all odd values of t , if n
Godwin [6
and 2.
I
I
=3
and 4 are given below.
Table I.
Here
~3(n,k)
A --
k
=n
- 1
2
2A
3
4
3A
2B + 2C
l
-5A' +
l + 5B2
;m
5
4:rr
1/2
k=n-2
1
= ----",,..,..,.
2 j2 :rr3/2
1
0
l2A - 6B l - 6c
2OA. - lOBI - lOC
Hence we can exactly
5 and all even values of t if
B
5
t + 4.
have given tables of exact moments ~t for
The corresponding tables for t
=n
k
and Jones ,[11
~
~
B
2
=
15
2:rr3/2
0
C
= 1~/2
2rc
arc tan (/2)
11
Here
Table II
1l4(n,k)
k
n
2
3
3
3+a
b
=
3-2a
4
3+2a
5
3+b4<c
3+10a-4b-4c
3-20a+6b+6c
6
3 -56.+ 3b+ 3c
3+ 25a-9b-9c
. 3-20§.1: 6b+ 6c
65. arc tan{
n" 2
" rc
/5/3)
3-2a
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[) I
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•
[2
I
Bechhofer, Robert E., "A single-sample multiple decision procedure for ranking means of normal populations with known variances,"
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[) I
Bechhofer, R. E., Dunnett, C. \L and Sobel, M., "A two-sample multiple decision procedure for ranking means of normal populations
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[4
J
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•
•
••
.e
[) I
Cole, R. H., "Relations between moments of order statistics," Ann. Math.Stat •
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[6
I
Godwin, H. J., "Some low moments of order statistics," Ann. Math. Stat •
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[) I
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[) 1
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[11
I
Jones, H. L., "Exact lower moments of order statistics in small samples
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•e
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I
Mosteller, Frederick, "On some useful inefficient statistics," Ann. Math.
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..
[17
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e
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[)8
I
Ruben, H., liOn the moments of' order statistics in samples from normal populations," Biometrika, Vol. 41 (1954), pp. 200-227.
[19
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Salzer, H. E.; Zucker, Ruthi and Capuano, Ruth, IlTable of the zeros and
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I
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1:21
I
Szego, Gabor, Orthogonal Polynomials, New York City, American Mathematical
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•
Teichroew, Daniel, Probabilities Associated with Order Statistics in Samples
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•
e