ON INCOMPLETE U-STATISTICS HAVING MINIMUM VARIANCE
by
Alan J. Lee
University of Auckland
and
University of North Carolina at Chapel Hill
Summary
Let U be an incomplete
U~statistic
of order k -- that is to sayan
ari thmetic mean of m quantities g (X.1 , ... ,X.1 ) where Xl"'" Xn is a
1
k
random sample from some distribution, g is a function sYmmetric in its k
arguments, and the sum is taken over m k-subsets (il, ... ,i ) of
k
{l, ... ,n}.
The problem of how to choose the m subsets to make the
variance of U a minimum is discussed.
Some results on the asymptotic
properties of U are given.
Key Words and Phrases:
incomplete U-statistic, design, balanced incomplete
blocks, consistent estimation.
The work of this author was supported by the Air Force Office of
Scientific Research under Grant AFOSR-80-0080.
2
1.
Introduction.
Let Xl"",X
be a random sample from a distribution with d.f. F,
n
where F is a member of a class F of dfs. Let 8 be a parameter such that
e=
8 (F)
== E [g (Xl'
... ,~) ] for all F
where g is symmetric in its k arguments.
8 (F)
= foo
F
(1)
In this case the functional
k
... J~oo g (xl' .•• , xk)
under certain conditions on
E
IT
. 1
1=
dF (x.) is said to be regular, and
1
F, the minimum variance unbiased estimator of
8(F) is the complete U-statistic
I
g(X.1 , .•• ,X.1 )
1
where the sum is taken over all
(~)
(2)
k
k-subsets of {I, ... , n} .
The amount of computation required to calculate (2) may be excessive
if nand k are large.
Because of the high degree of dependence between
the terms of (2), it appears that discarding some of the terms involved
in the U-statistic will not appreciably inflate the variance.
We are
thus led to consider incomplete U-statistics of the form
U
m
=
1
m
I
g(X. , ... ,x.
1
1
1
)
(3)
k
where the sum now extends overm k-subsets chosen in some manner.
Such
incomplete U-statistics have been studied by Blom (1976), Brown and
Kildea (1978), and Lee (1979).
In the present paper we address the question of how to choose the
sets defining (3) to make the variance a minimum.
We also discuss
asymptotic properties of sequences of optimal incomplete U-statistics.
3
5ince the variance of an incomplete U-statistic is always greater
than that of the corresponding complete statistic, one is trading off
efficiency against simple computation.
However, in some cases it is
possible to achieve considerable savings in computation coupled with a
negligible or zero decrease in asymptotic relative efficiency.
to this point in
2.
§4
We return
.
The variance of an incomplete U-statistic.
Following Hoeffding (1948)., define functions
gc(xl, ... ,xc ) = E[g(x1 , ... ,x c ' Xc+l, ... ,Xk )] for c = 1,2, ... ,k and
(J
2
c
= Var(g (Xl, ... ,X )).
c
c ..
Let 5.; j = l, ... ,m, be m k-subsets of
J
{I,2, ..... n} and let gj = g(Xil, ... ,x
ik
1
) where 5 j = {il, ..• ,i }.
k
The
m
U-statistic based on the 5. is Um -- ill. L\' gJ. and we call the sets 5.
J
J
J=l
the design of the U-statistic.The design is conveniently described by
the nxm incidence matrix N whose i-j element is 1 if i
€
5. and zero
J
otherwise.
From Blom (1976) the variance of U is given by
m
1
k
2
Var U = 2" I f (J
m
m c=l c c
(4)
where f
is the number of pairs of sets 5., 5., having exactly c
c
J J
elements in common. fc
is the number of elements of ·
N'N equal to c
If we assume the sets 5
j
to be distinct, then f
an alternative expression for the variance.
k
= m.
We now. develop
Let a(i , ... , i.Q) be the
1
number of sets that contain the set {il, •.. ,i~}, so that
4
m
= L
j=l
where N = (n .. ).
n ..... n ..
1 13
1.1/,3
Define for '\) = 1, .•• , k
1)-
A.I/,
=
B.I/,
=
Define the Stirling numbers S ('\)) and S (lJ) of the first and second kind
lJ
'\)
by
'\)
= I
'\)
x
s(lJ)
'\)
lJ=l
x(x-1) ... (x-lJ+1)
and
x(x-1) ... (x-lJ+1)
Then
n
[
n
m
I ... I
i ::; 1
1
n
i
I
i =1 j =1
'\)
n
m
I
I=1 .... i I= 1 j=l
1
'\)
r
n ..... n . .
1 13
1'\)3
=
'\)
I
lJ=l
11!
s(lJ) A
'\)
lJ
(5)
'\)
n. . . .. n. . = I lJ! s(lJ)
B
'\)
1 3
lJ
l'\)J
1
lJ=l
But the left-hand side of (5) and (6) are respectively equal to
(6)
5
(7)
(8)
Multiplying each side of (7) and (8) by
s~V)) summing over
and using the identity
yields from (7)
k
5I,!
A~
=
I
c=l
k
=
f
P
c v=l
V
sCV)c }
~
I f c(c-l) •.• (c-5I,+1)
c=l c
V from 1 to
~
6
and so
(9)
Similarly from (8)
~I Bn
~
=m I
v=l
N
SVkV - m k(k-l) •.. (k-~+l)
~
-
and so
(10)
Finally multiplying ~) by (-1.) ~-V
. ~ from
. ~~)
v' summing over
using the identity
~~V (-l)~-vr~l(~·
=
f
V
(~)Ocv
=
I (-l)~-v(~JA
~=V
V
(11)
~
m
where d~ =
I
(iI
(_l)~-V(~)Al
~l
~=V
~
...!-U.
2
m
~ (-1) ~-c (c~)O'2c·
cL
c
to k and
we obtain
so by (4)
Var U =
V
V
7
The quantities d.R, are non-negative by Lemma 5.1 of Hoeffding (1948),
so Var U is minimized by designs having all A.R, a minimum.
m
In general, finding designs which simultaneously minimize the A.R, is
a difficult problem in 0-1 programming.
However, in certain situations
we may be able to recognize minimum A.R, by simple arguments.
particular, note that A.R, is the sum of squares of
integers a(i l ,
,i.R,) whose sum is
which the a(i ,
l
B.R,
= m(~).
If a
non-negative
de~ign
exists for
,i.R,) are all equal, then A.R, will be a minimum.
necessary condition for this is that
Alternatively, if
m[~)
<
(~)
m(~J/[~)
=
B.R,'
A
is a positive integer.
and a design exists for which the
a(il, ... ,i.R,) are all zeroes and ones, then
zero, and so A.R,
[~)
In
m(~)
are unity and the rest
Cl early for any design A.R, ;c; B.R,' so A.R,. is
minimized if the a(il, ... ,i.R,) are zeroes and ones.
Moreover if for any
v;c; 2, a(il, ... ,i ) is equal to zero or one for all il, ... ,i ,then A.R,
v
v
attains its minimum for .R, ;c;
v.
To see this, note that if a(il, ... ,i ) is zero or one for all
v
subsets iI' ... , i
.R,
;C;
V
and so A.R,
v
then necessarily a(i , ... , i.R,) is zero or one for
l
=
B.R, and so attains a minimum.
We now turn to the description of designs for which the A.R, have one
of the two properties above.
3.
Designs having minimum variance.
3.1.
Designs based on tacticaZ configurations.
A class of designs
for which the A.R, are sums of squares ·of equal terms are those based on
tactical configurations.
A tactical configuration (t.c.) C(k,.R"o,n) is a
system of m k-subsets of· {I, ... ,n} such that every .R,-subset of· {I, ... ,n}
8
is contained in exactly 0 of the m subsets of the systenl.
A necessary
condition for the existence of a t.c. is that the quantities
oh
are integers for h
= 0 [n-h]
R--h I [k-h)
R--h
= O,l, ... ,R--l.
C(k,h,oh,n) t.c. for h
If such a t.c. exists it is also a
= 1,2, .•• ,R--l. The number m of sets is given by
These facts may be found for example in Raghavarao (1971).
Suppose now that a C(k, k-l , 0, n) tactical configuration exists.
Then the incomplete U-statistic based on this t.C. has minimum variance,
for by definition everyR--subset of· {I, ... ,n} is contained in oRsubsetsS
of the system, and so each AR- is a sum of squares of equal
j
terms and so is a minimum.
Special cases.
a) k = 2.
A C(2,1,0, n) configuratioriis simply a system for which
each element i is contained in the same number 0 of sets S .•
Designs
(for any k) having this property are termed balanced by Blom.
The
J
balanced designs for k
=2
have been considered by Brown and Kildea
(1978), who prove the asymptotic normality of U-statistics based on them.
b) k = 3.
A C(3,2,o,n} t.c. is just a balanced incomplete block
design for n varieties with m =. 0 [~] I 3 blocks of 3 plots each.
pair of indices (varieties) occurs in A
Each
= 0 sets (blocks), and the
design is balanced in the sense of Blom with each index appearing in
9
r = o(n-1)/2 of the sets S ..
J
The variance of the incomplete U-statistic
is given by
Var U
m
A sequence of designs having m = 0
triple systems.
[~2]
are those based on Steiner
See e.g. Raghavarao (1971) p. 86.
These areBIBD's with
parameters k = 3 and
n
= 6t
+ 3
m = (3t+l) (2t+l)
r
= 3t
n
= 6t
+ 1
m = t(6t+l)
r
= 3t
A= 1
+ 1
or
and exist for every positive integer t.
based on them are
A= 1
The variances of U-statistics
(9tC1i+C1~)/(3t+1)(2t+1) and (3(3t-1)C1i+C1~)/t(6t+1)
respectively.
c) k= 4.
and have m =
(mod 6).
3.2.
C(4 13 11 1n) configurations are called quadruple systems
(~)/4
subsets.
They exist (Hanani (1960)) when n = 2 or 4
Little is known about t.c.'s for which k > 4.
other designs based on taatiaaZ aonfigU1'ations.
a C(k 1\)111 n) configuration exists for V
~
contained in at most 1 of the Sj
A~
Since the design is a
minimized for ~ <
V
C(k1J(;1o~1n)
and so
t.c.
2.
Suppose that
Then any V- set is
is minimized for
for 1
:S ~
<
V
the
A~
~ ~ V.
are also
and so the U-statistic has minimum variance.
In
particular BIBD designs for arbitrary k 1n 1m are of this type if A (the
number of blocks containing any pair of varieties) is unity.
10
3.3.
Designs based on partiaUy balanaed inaomplete bloaks.
We
consider designs based on two associate classes with n varieties, k plots
per block and m blocks.
Let n , n '
l 2
~\,
A be the parameters of the
2
association scheme, so that every variety has n
n
2
l
first associates and
second associates; any pair. of varieties that are i th associates
occurs in \
blocks, i = 1,2.
Suppose Al = 1 and A = 0 (or Al = 0
2
and A = 1); any pair of varieties occurs in at most one block and so
2
A is a minimum.
Z
also a minimum.
minimum for 9.,
~
Since the PBIB is balanced in the Blom sense, Al is
Since the a(i , i ) are zeroes and ones, A9., is also a
l 2
2.
Thus the U-statistic based on a PBIB with two
associate classes has minimum variance if Al = 1 and A = O.
2
3.4.
Designs based on ayaUa permutations..
We now consider a class
of balanced designs, with m = nKfor some integer K, that have the
property that the off-diagonal elements of the matrix NN' are zeroes and
ones.
Since A is the sum of squares of the upper off-diagonal elements
2
of NN', such designs correspond to minimum variance U-statistics.
first consider the case K = 1.
We
Then m = nand r = k, and for any
balanced design N is square with row and column sums equal to k.
It
follows (see e.g. Raghavarao (1971), p. 107) that N can be written
k
N=
I
P9.,
9.,=1
where P9., is a permutation matrix, 1. e. one whose i-j
form O.
1,
element is of the
.(.) for some permutation p of the integers· {1, .•. ,nL
p J
Conversely, any set ofk permutations Pl"'o, Pk will generate a balanced
design, provided P9., (j);t P9., (j) for distinct 9.,1,9.,2 and j = l, ... ,n.
1
2
11
Now let q be the cyclic permutation q(j) = (j+1) (mod n), and for
aR,
~
Let PR, be the permutation
k
matrix corresponding to PR,' and set N =
PR,' We note that, provided
R,=1
the aR, are distinct, N is the incidence matrix of a balanced design.
k
k
is the permutation matrix
Also NN' =
PR, P
where PR, P
R, =1 R, =1
1 2
1 2
1
2
aR, -aR, .
integers aR,' 0
aR, < n, set PR, = q
•
I
I
1
I
1
corresponding to the permutation q
2
1
Now call the elements of an nxn matrix for which i - j = c (mod n)
1
the c-diagona1. of the matrix.
The matrix PR, P
has ones on its
1 2
diagonal and zero elsewhere. Thus, provided the quantities
aR, - aR,
2
1
aR, - aR, . for R,1' R,2 = 1, .•. , k are distinct (mod n) when they are not
2
1
zero, NN' has its off-diagonal elements zero or one, and hence N is the
incidence matrix of a minimum variance
gives sui table values for the
aR,
U~statistic.
The table below
for different k values.
TABLE 1
Values of a . for different k values. The range of n yielding minimum
variance u-s\atistics appears in parentheses.
e,
R, = 3
R, = 4
R,= 1
R, = 2
k = 2
0
1
k = 3
0
1
3
k = 4
0
1
4
6
k = 5
0
1
4
9
k = 5
0
1
4
14
R, = 5
(n
~
3)
(n
~
7)
(n
~
13)
11
(n
~
23)
16
(n = 21)
12
For K > 1, it may be possible to construct suitable incidence
matrices of the form N = [N l : N2 : .•• : NK ] where each N.Q,'
is of the above form.
Then NN'
=
.Q,
= 1, ••• , K ,
NlNi + ••• + NKNi: will have its
off-diagonal elements zero or one if the matrices N.Q,NI have their
non-zero diagonals all distinct.
choose N1 based on
a..Q,
For example if k :: 3 and K = 2 we can
values 0,1,3 and N based on 0,4,9.
2
resulting N has distinct non-zero diagonals for n
~
The
19 .and so is a
design for a minimum variance U-statistic.
4.
Asymptotic results.
First we comment on the asymptotic relative efficiency of the
designs above.
Let U
m
n
be a sequence of U-statistics based on nxm
n
incidence matrices N which have the property that A =
n
2
m(~)
or
equivalently that the off-diagonal elements of N N' are zeroes and ones
n n
and are balanced.
Then, dropping subscripts and assuming
Var U
m
where
f
1
= kI
.Q,=1
I
k C_1).Q,-v
C_1).Q,-v ( .Q,) A
v .Q, = .Q,=2
since
v
~
2 •
cri
> 0,
13
Thus
f
1
(~) [~)
= m
+
rmk - mk
= mk(r-l)
so
The variance of the complete U-statistic is
Var Unc
2(1)n
1 2
= -k
n °1
+ 0 .;....
so the asymptotic relative efficiency of the complete versus the
incomplete statistic is
lim
= lim
n 400
n 400
2
2 2 (r-l)
ko
- - +kOk
--
1
Stipposethat ~ -+
n
00
so that r -+
00 •
r
r
Then the ARE is unity, and if ~ -+ b,
n
2
2
2
say, then r -+ bk and the ARE is bkoll(k(bk-l)Ol + Ok) < 1.
For example,
a sequence of designs having the former property are those for k = 3
based on the Steiner triple systems, and a sequence having the latter
property are those for k=3, m=n based, on cyclic permutations with b = 1
and an ARE of k
°1 / (k(k-l)Ol2 + Ok)2 .
2 2
For results on asymptotic normality, once again consider a sequence
of designs based on incidence matrices N of the type described above.
n
14
We distinguish two cases:
.
(i) r
n
-+
(ii) r
00 ;
2
2
=
bk with b finite.
2
2
k0
k(r-l)
+ ---!.
and so
-+
ci
then n Var Um =iil (k(r-nOl + Ok)
r
1
r'
C
2
C
h
U lS
· 1e t e
•
. n Var U = k 2 = l'1m V·ar U were
11m
t h e correspond
lng 'comp
1
m
n
n
n
statistic. It follows from §S of Blom (1976) that the asymptotic
If r
-+
00,
°
m(U~ - 8)
distributions of
When r
-+
kb <
00,
and meUm - 8) coincide and are N(O ,k20i).
n
2
2
then m Var U
converges to k(bk-l)Ol + Ok and a
n
m
n
trivial extension of the argument of Brown and Ki1dea (1978), Theorem 1,
shows that rm(U - 8) converges in distribution to N(O, k (bk-l)
m
n
oi
+
O~) •
To construct confidence intervals for 8, we need a consistent
est ~mator o.f k 20 l2 or k (bk
.L
-
°
1) 1
2 + Ok2 as approprla
. t e.
Define for
i=l, ... ,n
where U(l") = _1_
g.J and the sum
m-r I(.)
1
containing i.
I (.)
1
is taken over. all sets S.J not
Then if '"
W' = (WI"'" W) and g = (gl"'" g ), W = Ng .
n
N
m '"
1'.1
"
Consistent estlmators of k
statistic
I (W.1 - w) 2
°1
2 2
2
2
and k(kb-1)01 + Ok may be based on the
1..
n
n
r
W.. Similar statistics are
. 1 1
1=
employed in the complete U-statistic case for consistent estimation of
k
2 2
°1 ,
where W =
.
Consider
W' (I- 1.. J )W
'" ·n n n '"
= g (N
where I
n
and J
respectively.
n
I
2
k
N- -. J ) g
n
m '"
are square identity matrices and matrices of 1 I S
The covariance matrix of the vector
~, TI
say, has elements
15
7T ••
lJ
= 0 c2
Thus
where c is the i-j element of N'N.
=
k2
trace IT(·N'N-- J )
n
m
2
+e 2 '"1 1 (N'N-L
n
J )1
m'"
(12)
2
k
= mk(r-l) (1- n
)0
2
.
k
2
+ mk (1- n )Ok
1
1\
Now consider the case r +
00 •
The estimator
asymptotically unbiased estimator of
lim E(~
mr
n
S
00
=
k20~, since
L (W.w)2) = lim(k 2 (r-l) i + L
1
r
1 r
2 "
0k2 )
.
=
n
k
L (W. _ W) 2 is an
mr i=l 1
k 20 2 •
It is also
1
consistent; to prove this (under the assumption Elg(x , ... ,x
l
k
)1 4
< (0)
write
k
mr
n
L
i=l
(W. _ W) 2
1
=
~
mr
N IN
£
£
_
L
[ ~ ]2
nmr j ~ 1 gj
(13)
m
L
j=l
where
L(1)
denotes the sum over all sets Sj having one element in
Since U converges to e in probability, the last term in (13)
m
m
2 2
converges to _k e in probability. Also 1.. L g~ is an incomplete
m j=l J
U-statistic based on the kernel
and so converges in probability to
common.
i
E(g2)
= O~
+
~2.
Thus the second term converges in probability to zero.
16
For the first term, consider
var(~
L(l)
mr
g.g.,)
J J
where the sum is taken over all j, j , ,51,,51,'
and St () St'
such that the sets S. () S.,
J
each have exactly one element.
J
If (Sj U Sj ,) () (StU St') =
<P
the corresponding term in the sum is zero, since gjgj' and gtgt' are
independent in this case.
The number of terms for which this is not the
3
case is less than m. rk. (2k-l) r.kr = 0 (mr ) and so
3
2
k
k
3
k
Var(-· L(l) g.g.,)
"""22
O(mr )
0 (1) converges to zero.
mr
. J J
mr
=
k
E(mr
=n
Since
kf l
2
2
2 (r-l)
2
mr (a l + e) = k
r (a l + 8) it follows that the
2
2
first term of. (13) converges in probability to k (ai + 8 ) and so (13)
L(1) gjgj ,)
=
2 2
converges in probability to k a •
1
Thus when r
2 :2
is a consistent estimator of S = k a •
l
+
00,
S::
~
I
mr '-1
lIt follows that when r +
(W. _ W)2
1
00
;fS:ln [um-e} has asymptotically a normal distribution with mean zero and
variance unity.
For the case r
.!.
L(W.w)2;
m
1
~
L (W i - w)2
+
kb < 00, the estimate must be modified.
Consider
2
2
+ ka
l
k
so
is not an asymptotically unbiased estimate of k(bk-l)ai +
a~ .
from (12) we obtain
lim E .!.
.
m
n
L(W.w)2
1
= k(kb-l)a
Using a decomposition similar to (13) we obtain·
.!.
L(W.
m
. 1
W) 2 =
~ L(I) gJ.gJ"
+
~m
I
j=l
g:
J
which using the arguments above converges in probability to
2 .2
2
2
2 2
k(kb-l) (a + 8 ) + k(a + 8 ) - k be
l
k
= k(kb-l)a 2
+
2
ka
k
17
To adjust the estimate, consider
1
m
-m-1
I
j=l
(g.- g)
2
J
g' (I - 1:. J )g
= _l:m-l,.., m m m,..,
We have
~
1
E (m-l)
.
_~
m
)-1
.~ (gj-g)
11·
= -m-1 trace II (I m- -m J m)
= (i
k
k(r-l)
m-1
2
Ci l
m 2
1
2
1
m .
_21
and L
(g.- g) = g. - -1 U converges in probability to
m-1 j=l J
m-1 j=l)
mm
2
Ci .
Thus -.!-l
I (g.) - g) 2 is an asymptotically unbiased and consistent
m- .
k
r
estimator of Ci~.
It follows from the above that
B=
k
I
mr 1'=1
(W. - W) 2
1
k -1
I
- m-l j=l
(g). _ g) 2
is an asymptotically unbiased and consistent estimator of k(kb-l)q~
and so /
S-l n (Urn- eJ is
asymptotically N(O, 1) •
Referenc·es
Blom, G. (1976).
Some properties of incomplete U-statistics.
Biometrika 63, 573-580.
Brown, B.M. and Kildea, D.G. (1978). ReducedU-statistics and the
Hodges-Lehmann estim~tor. Ann. Statist. 6, 828-835.
+
Ci~
18
Hanani, H. (1960).
On quadruple systems.
Can. J. Math. 12, 145-157.
Hoeffding, W. (1948). A class of statistics with asymptotically normal
distribution., . Ann. Math. statist. 19,293-325.
Lee, A.J. (1979). On the asymptotic distribution of certain incomplete
U-statistics. Institute of Statistias MimeoSeries #1259,
University of North Carolina at Chapel Hill.
Raghavarao, D. (1971). Construations and Combinatorial Problems in
E:x:perimenml Design . . Wiley: New York.
---,,-----------------
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (WI't'1I IJulul'lIt"rod)
r
READ INSTRUCTIONS
BEFORE COMPLETING FORM
REPORT DOCUMENTAriON PAGE
t. REPORT NUMBER
GOVT ACCESSION NO.
4. TITLE (and Subtitle)
1.
RFCIPIEtlT'S CATALOG NUMBER
5.
TYPE OF REPORT Ilo PERIOD COVERED
TECHNICAL
On Incomplete U-Statistics Having Minimum
Variance
6.
PERFORMING O~G. REPORT NUMBER
Mimeo Series No. 1283
7.
8.
AUTHOR(s)
Alan J. Lee
9.
11.
CONTRACT OR GRANT NUMBER(s)
Grant AFOSR-80-0080
10. PROGRAM ELEMENT. PROJECT, TASK
PERFORMING ORGANIZATION NAME AND ADDRESS
AREA 8< WORK UNIT NUMBERS
12.
CONTROLLING OFFICE NAME AND ADDRESS
Air Force Office of Scientific Research
Bolling Air Force Base
Washington, DC 20332
14, MONITORING AGE.NCY NAME Ilo ADDRESS(1f dilf",,,,,' Irom
COli troll/nil
REPORT DATE
April 1980
13.
NUMBUi OF PAGES
15.
SECURITY CLASS. (of thl •• report)
18
Olflce)
t
UNCLASSIFIED
'1"5,;."
16,
DECL-A"SSI-FICATlON/DOWNGRADINoSCHEDULE
DISTRIBUTION STATEMENT (of this Report)
Approved for Public Release -- Distribution Unlimited
A
17.
DISTRIBUTION STATEMENT (of the abslrect entMed In Block 20, If dlfferenl from Report)
18.
SUPPLEMENTARY NOTES
19.
KEY WORDS (Continue
Oil
reverse side If necessary and IdentIfy by block nt/mber)
incomplete U-statistic, design, balanced incomplete blocks, consistent
estimation
20.
ABSTRACT (Continue on reverse side If necessery end IdentIfy by block nt/mber)
Let U be an incomplete U-statistic of order k
--
that is to say an
arithmetic mean of m quantities g(X. , ... , X. ) where Xl' •.. , X is a random
1
1
n
1
k
sample from some distribution, g is a function symmetric in its k arguments,
t
and the sum is" taken over m k-subsets (il, .. ·,i )
k
DO
FORM
, JAN 73
1473
EDITION OF 1 NOV 6515 OBSOLETE
of {r, ... ,n}.
The problem
::J . I
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (When Data E n t - = - - J
•
SECURITY CLASSIFICATION OF THIS PAGE(When Dill" EnlMftd)
200
UNCLASSIFIED
of how to choose the m subsets to make the variance of U a minimum is
discussed.
Some results on the asymptotic properties of U are given.
UNCLASSIFIED
SECURITY CLASSIFICATION OF
Tu'O
PAGE(When D"t" Entered)
•
,
j