r
UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, No Co
Mathematical Sciences Directorate
Air Force Office of Scientific Research
Washington 25, D. c.
AFOSR Report No.
""~i::
892
SOME PROPERTIES OF THE LE..L\.ST SQUARES ESTIMATOR
IN REGRESSION ANALYSIS WHEN THE
i
INDEPENDENT i VARIABLES ARE STOCHASTIC
by
P. K. Bhattacharya
University of North Carolina
June, 1961
Contract No. 49(638)-213
Qualified r~questor8'inay'obtain copies' of this report 'from 'the ASTIA
Document Service Center, Arlington Hall Station, Arlington 12, Virginia.
Department of Defense contractors must be established for ASTIA services,
or have their "need-to-know" certified by the cognizant military agency
or have their project or contract 4
#_-.' ..
~ .-JIII!t;.,
.''''''.'"
~,.
Institute of Statistics
.
Mimeograph Series No. 292
10
Introduction and Summary.
In the classical
linear estimation set-up, we have
E 1 = XQ
(1)
y t = (yl, ••• ,Y)
where
-
n
is a random vector whose components are uncorrelated
Gl= (G , Gl, ••• ,Q)
and have equal variance,
unknown
,
o
p
X is a matrix of
constants and
is a vector whose elements are
n rows and
p+l
columns,
n~p+l,
Plackett ~g7
which has full rank and whose elements are known constants.
gives a historical note on the least squares estimator
£,
of
for which the following property is well-known, --
e.
(I) Each component
J
of
e
is the uniformly minimum variance, unbiased,
linear (in y1s) estimator of the corresponding component
of
Q
j
G.
It can also be easily seen that
(II)
of
For a quadratic lOBS function for the estimator of each component
the least squares estimators have uniformly minimum risk among the class of
~,
all linear (in yt s ) estimators with bounded risk.
Lehmann and Hodges ~!7 have shown that if we do not restrict ourselves to
estimators which are linear in
y1s, then the least squares estimators have the
following weaker property, -(III)
If the loss in estimating the true vector
(~ - ~)I
(~ - ~),
Q
contains the sub-family
~
if there exists a number v
yi:S v, i = 1, ••• , n,
(y., ••• ,y)
n
by another vector
~
is
then the least squares estimator is minimax among the class
of all estimators of
Var
~
r-r'
J1 0
such that
rf{
and the family of distributions
of (Yl ,· ."Yn)
of all independent normal distributions
which satisfy (1)
for some
Q, and have
-
Var y.
~
= v,
i
of
= l, ••• ,n.
In many situations, however, (y, xl, ••• ,x p ) follows a (p+l) - variate distribution on which observations are made and the method of least squares is
2
applied to estimate the linear regression of
x - observations to be non-stochastic.
y on
xl, ••• ,x p ' regarding the
This problem differs from the classical
problem of linear estimation in two respects i
(a)
the model should be
Instead of (1)
ELl
I x_7 =
where the elements of the
(b)
X
X 0
,
are stochastic, and
matrix
the problem being that of estimating the entire regression function
rather than the coefficients of the linear regression individually, the loss
function should be defined in the space of all linear functions of
xl, ••• ,x p •
For reasons given in section 2, the loss in estimating the true regression
~(Xlj ••• 'Xp)
function
~(Xl' ••• 'Xp)
by another function
may be considered
to be of the form,
f
where F
L~(xl' ... ,X p ) - ~(xl'" .'Xp)_r
is the distribution function of
d F(x l ,.· .,x p ),
(xl, ••• ,Xp ).
For the above loss function, it 1s shown under certain conditions that if
the class of estimates which are linear in
y's
and have bounded risk is non-
empty, then the estimate obtained by the method of least squares belongs to this
class and has uniformly minimum risk in this class.
condition on
F(xl, ••• ,x p )
A necessary and sufficient
is obtained for this class to be non-empty, which
unfortunately is not easy to verify in particular cases and is
very simple situation.
violated in a
However, by a sequential modification of the sampling
scheme, this condition may always be satisfied at the cost of an arbitrarily
small increase in the expected sample size.
It is also shown under certain
further conditions on the family of admissible distributions that the least
squares
2.
esti~ator
is minimax in the
cl~es
of all estimators.
Formal statement of the problem.
Let
y, . xl' ••• 'xp
be real-valued random variables whose joint distribution
satisfies the following conditions, --
3
Condition (i).
The distribution
*(a)
function
for every non-null
a'
=
F
of
..
xl'$OO,x
is such that
p
(a ,al, •• a,a ), the set
9..
P
= {(xl,OGO,x);
+alxl+ooo+a
x = 0)
p ao
.'
pp
S
~
has probability measure zero ..
(b)
= E(xjx.,)
.J
~j"
.
J
is finite, j,j'
= O,l,o.o,p,
E~ylxl, ••• ,xp-7 is a linear function of
Condition (ii).
x
o.
-
1.
xl'~."'xp, say
~(Xl''''''X
. p ) = Q0 + ~\ xl + ...+ 0px p 0
V~yIXl, ••• ,xp-7 = 0 2 < aD , which does not depend on
Condition (iii).
(xl,···,Xp)o
(Yi'Xli, ••• ,x pi )' i=l, ••• ,n, n ~ p+l,
on (y,xl,.o.,Xp)o
The problem is to estimate the regression function
~(Xl' ••• ,x p ) =
oo + 0lx l + ....+ 0px P ,
or in other words the row vector
volved in estimating
(2)
W(~,~) =
f
~
is,
P
L:
L:
j=o j
=
by
~
(0, 0 , •• 0,0 ), where the loss ino
1
p
Of =
2
.L(O + Qlx + •••+0 x )-(13 +l3 x + •••+13 x )7 dF(x , ••• ,x )
l l
l
l
o
pp
0
PP-P
p
=
are mutually independent observations
(~f
f=O
_ ~ f) M (~ _ ~),
where
1
~l·"
=
M
~p
•••
~Op
~p
It follows from condition
o <
•••
~pp
•
M is positive definite, and
(i) that
W (0, 13)
<
aD
*This condition is satisfied if
xj
=x
j
or if
xl' ••• 'xp are
x
has a continuous distributi9n and
sin x, sin 2x, ••• , cos x, cos 2x, ••••
4
for all
~
~
g.
The loss function (2)
is motivated by the following consideration.
Suppose we are required to predict the value of
observation made on
(xl, ••• ,X )
p
y
associated to a random
subject to a quadratic loss.
If the true
regression function
=
were known, out prediction rule would be
Y (xl''' .,X p ) =
o + Ql xl + •••+ Qp x p '
Q
and the risk of the procedure would be
2
cr.
If however, we use the prediction
rule
y' (xl'···,x p
) = ~ +o
~l xl +p
•••+ ~ x,
P
2
cr + W (~, ~).
then the risk would be
3.
Optimum property of the least
8:\jj;U8JU38
estimator in the class of linGsr
estimators with bounded risk.
In this section we shall restrict our attend ion only to those procedures
which are linear in
such procedures by
y's
~
and mve bounded risk.
=
t
-p+lxl
(3)
x =
L
p+lxn
is a bounded function of
function of
Then an estimator
l'
t
Let us denote the class of all
€
~
1
if and only if
'l.
nxl
Qo' Ql, ••• ,Qp' where each element
Xll""'Xln, ••• ,xpl, ••• ,Xpn'
·.. xp1
1
X11
1
x
x
12 • •• p2
••••
1
x1n
·.. xpn
and
'l.'
=
£ji
of
(Y1, ••• ,Yn ).
L
Let
is a
5
Then the estimator obtained by the method of least squares is,
Since
p+l
t*
=
n ~ p+l,
it
follows from condition (ia) that the matrix
with probability
1
and therefore
t*
X has rank
can be uniquely determined for al-
most all samples.
E Ylx ~U(x,y)_7
In what follows,
stands for
We shall first show that if an estimator
!
€
e
E
l'
X, the conditional expectation of each component of !
equal to the corresponding component of
For any !
(j
€
£,
it follows from (3)
l'
whatever
y ~U(x,y)
then for almost all
I(X
where
h
(t j )
jj
i~l
=
may be.
9
that
2 ji (9o + 91 Xli + •••+ Q x 0)
p p~
are funct ions of
,
in its
such a matrix
jth
row and
Let
j'th column.
H
=
P
E
h
j + j'=o
JJ
Q
o
be a (p+1) x (p'H)
To every !
€
~
l'
o ,
QJ.t
say,
matrix which has
there corresponds
H, and to show this correspondence, we shall use the notation
Ht • Thus if !e:{;l'
(4)
X.
X should be
given
n
E
Ix_7.
E~Jx(!) -
then
=
9
Ht
~
We now prove
Lemma L
If!
for almost all
Proof.
€
.-g l'
then
X.
By virtue of
(4) it will be enough to show that if
P~Ht ~
then ! ,
bounded.
e 1.
Let
We have
!
0_7 > 0,
satisfy (3); then we shall show that
r (9, t)
is un-
6
If
P~Ht ~
Q7 >
0, some element of
Let that element be one in the
Then for
Q' H'
P~g(X)~
Q7
=
Q
-
€
j ~ jo) •
for
J
g(X)
Jo
P~g(X)
and
> 0_7> 0,
since
M is positive definiteo
such that
p(~g(X)
> ~7 > o.
o
and for any given
A.
Jo
2
Q.
A.,
r(.2, ~) ~
in
1
e> 0
pee) =
Then for
=
M H ~
t
t
Hence there exists
9
Ht • Let
Q € A. ,
Jo
where
is non-zero with positive probability.
joth column of
=0
Qj
H
t
EX
c,
1.2 '
t
H
M H
-
-
~J ~ g~ e pee),
t
r(~,!)
0
can be made greater than
c
by choosing
with
•
This completes the proof.
The following corollary is immediate.
Corollary.
(5)
r (~ ,
If
!)
!
=
€
e
2
(J
l' then
I;
I;
j=o
where
R...
J~
n
p
p
Ex
j'=O
Il jj I
are the elements of the matrix
I;
R. .. R.. , •
J~
J ~
i=l
L through which
t
is defined
and they satisfy
(6)
n
n
I;
I;
i=l
i=l
for almost all
X.
R.
ji
X jti
=0
for
j
4 jt = O,l, ••• ,p
7
It follows from the above corollary that for
~.
depend on
!
€
~l
r(2,!)
does not
'
Therefore 1 if we minimize the right side of
(5) with respect
A
to
£ .. ,
J~
subject to the conditions in
s
(6), the resulting matrix L will de-
fine an estimator
for '\>lhich
A
r(~,
for all
Q
~
!)
.
r(~
!)
1
and for arbitrary !
e
€
1.
A
Now the equations giving L
A
(7)
ML
(8)
b..
=
A
=
LX
are
X'
I
where~ is a (p+l) x (p+l)
a.e.,
(p+l) x (p+l)
unit matrix.
A
I
is the
(7) and (8) we get
Solving
X' ,
=
L
matrix of constants and
and the resulting estimator
A
= (X'X)-l X'
t
where
t*
I
=!*,
is the estimator obtained by the method of least squares.
But we still do not know whether
seen that unless
Theorem
a.e.,
1.
-e
1
is empty 1
If {J 1
!*
€
!*
-e 1.
€~ 1. However, it can be easily
We thus l:B.ve
is non-empty 1 then the least squares estimator
!*
€
e
l
and
r(~
for all
1
!*)
~
r(~
1
g and for arbitrary !
If we denote by
(;2
!)
€
~ 1 with a strict inequality holding if
the class of estimators which are linear in
then the following corollary is immediate from the fact that
constant for all
Q.
r(~
1
!*)
y's,
is a
'
8
If
Corollary.
tf/ l
is
the unique minimax
non-empty, then the least squares estimator
estimator in
The optimum property of
t*
is
g 2'
thus depends on the non-emptiness of ~ 1
t*
which can be characterized in terms of F(x1,o •• ,x). We have seen that ~l
.
non-empty if and only i f
r(~,
Let us denote by B the
(p+l) x (p+l)
!*)
p
is finite.
Clearly,
matrix
of which the
is the expectation of the corresponding element in
A,
is
(j,j')th, element
(X'X)-l and for any matrix
let us call the largest of the absolute values of its eieoents, the noro
{fl.AU
of
€Iap'by
is,
A
9
Condition (ic).
Then a necessary and sufficient condition for
/I B
II <
1 being non-
CD.
Thus if ~he joint distribution of
(y, Xl"'"
along with (ia), (ib), (ii) and (iii),
is non-empty"
e
Xp )
satisfies condition
then we can delete the phrase
(ic)
"if
g1
in the statements of theorem 1 and its corollary.
4. Minimax property of t* in the class of all estimators.
Let F be the unknown marginal
~
distribution of
be the family of admissible distributions of
F as the marginal distribution of
(Xl' •• o, Xp )
(y, xl, ••• ,Xp )
and let
which have
(xl, ••• ,Xp )'
In this section we assume
and (ic)
and the familY~ to satisfy
F to satisfy conditions (ia), (ib)
condition (ii) and
Condition (iv).
v
=
where
Sup
G
€% Xl'
VG f
variance of
Condition (v).
~ includes
Sup
o ••
,xp
V
G
LyIXl ,u.,xp-7 <
Y I xl'" .,xp
y given
the class
_7
co,
stands for the conditional
xl,o •• ,x p under G.
%
0
of all G obtained by taking
the product of the distribution F
of
(Xl,.o.,X p ) with a
9
which is
conditional distribution of
~
normal with mean satisfying (ii) with some
variance v
e
Under - these conditions we shall prove that
Q.
and with
!~
is a minimax estimate for
We shall require the following lemma to prove the minimax property.
Lemma
2.
Let A(z)
space of all kxk
II A(z )-1
II
be a mapping from an arbitrary space Z to the
non-singular matrices such that
are both bounded.
Let
be a fixed
U
sequence of 'real numbers converging to zero.
(a)
Det ~A(z) + bm
(b)
These
0
A{z)
/I
and
kxk me tr ix and (b m) a
Then,
converges to Det A(z) uniformly in z.
exists an integer -mo such that
form> m
-
Q7
II
and for all
z
~A(z) + bm ~7-1 exists
Zo
€
IIL-A(z) + b ~7-1 - A(z)-l II converges to zero uniformly in z, where
m
~A(z) + bm ~7-1 is defined to be the matrix each of whose elements is
(c)
max
f
Sup
II A(z) II,
Sup
zeZ
Proof.
zeZ
Suppose
max
L
II
A(z)-l IL7 + 1 if Det LA(z) + bm TJ..7
Sup
II
II,
A(z)
Z€Z
Sup
zeZ
II
A(z )-1 II,
II
U
IL7
=
= o.
c•
Then (a) follows from the fact that
k
IDet LA(z) + bm TJ..7 - Det A(z) I ~ Ibm' c • kt ,if Ibm' ~ 1.
k
k
Since IDet A(z)-ll ~ c
kt , IDet A(z) I ~ l/c • kt • Also, it follows
0
from (a) that there exists an integer m such that for m > m
0 - 0
k
k
Det A(z) - 1/2c • k~ ~ Det £A(z) + bm TJ..7 ~ Det A(z) + 1/2c
kt
0
for all
z e Z.
Hence for
m> m
-
IDet £A(z) + bm TJ..7 I ~ 1/2c
and therefore (b) follows.
(c)
is proved
$
and for all
0
k
0
z e Z,
k~ ,
soon as we apply (a) to the determinants of
(A(z) + bm U}, m = 1,2, ••• and all their cofactors.
10
Theorem
~.
Under conditions (ia), (ib), (ic), (ii), (iv) and (v), the
least squares estimator
timators
t* is
minimax for
Q
in the class of all es-
0
We shall first prove that
Proof.
Sup
G€1"o
for all
<
reG, t*)
-
Sup
G€%o
-
reG, t )
t.
Since G €~1t
instead of
is completely described by £,
0
reG, i)
for arbitrary i
variation over..e; 0
where £
we can write r(£, i)
corresponds to
is the same as variation over
£.
G,
and
So we shall show
that
r(£, i*) ~ Sup r(£, i )
Sup
g
for all
g
Suppose
t.
there exists
~
such that
1\
Sup
(10)
i) =
r(£,
Q
Sup r (£, 1*) - €, €
g
> o.
We shall argue up to a contradiction to (10) and (9) will follow.
Conditions (ib) and (ic) implies that
Hence we can find a constant
J
I
(X: lI(x'x)II
Denote the set
- E
tr~(X'x)-l ~7 I < €/~·
S c(€), II (x'x)-lll S c(€) }
(X: lI(xtx) II ~ c(~ ), II(X'X) -1 II
~
Consider the sequence {
d~m(£) =
c(€) such that
~7 i~l dF(xli,uo,Xpi )
trL(X'x)-l
E tr~(x'X)-l ~7 exists.
m
s c (€)}
by R(€).
} of a priori distributions of
(21tm)-(P+l)/2
exp
£ - ~£' ~7
~
j=o
Since the joint probability differential of (y, xl, ••• ,X p )
is
n
1t
i=l
where
n
dYi
1t
i=l
dF(Xli,···,X pi )
such that
g
d
Q
j
•
under G
o
11
and
£
corresponds to
!
f;:
r(Sm' !) =
£
=
>
G, we have
fl£-£)'M(~-i)gQ(rIX)
i.
X
-
fi£-!)tM(~-!)g£(rIX) d~m(~)_7
X
-Q
.
-y
r
£
r
m, since the hypotheses of lemma 2
and for all
X
€
R(e).
{Q _(XIX +! 1)-lX'y )'(X'X +!
m
m
Co"
m
Now for such large values of
m and for any
I) {Q-(X'X +! 1)-lX'y )_7
-
m
-
P
x
J(£
0
are ensured for
! 1)-1 exists for all
such that (X'X +
m
.
x expr-_ ~
>
dy . . 3\ dF(x 1i ,···,x i)
l
1= I P
J L J(~-!)' M(~-!)
/
and hence there exists
L
3\
1=1
£
for sufficiently large
m>m
o
n
i:1dYi
X€R(€ )
R(€)
n
ff fi£-!)'M(~-i)g£(rIX)d!;m(£)l
f·
= Const.
€
n
dy . . 3\ dF(x 1i , ••• ,xpi17 d!;m(~)
i=l
1 1=1
3\
f Jf
X€R(€)
X
n
3\
j=o
{£ - (XIX + ! 1)-1 X'y )
_(XIX + !: 1)-1 X'y ), M
m
-
m-
Q
x eXPL-
.L (Q _(XIX + ! 1)-1 Xly
m
2v-
-
)1 (XIX + ! I) (Q _(XIX +
m
-
P
X
3\
j=o
'1:-1
(since
M is positive definite)
! 1)-1 X'r )_7
m
12
•
• •
If
\.ie-fine
'w€'
t
-m
= (X'IX +!m I)-lX
= (XIX)-l
then
>
r(~
XiX
l
y
-
(X'X + ":!. 1)-'1
is'
m
other~ise,
A
, t)
m -
I f L- 1(~-.~)IM("Em-~)
XeR(€ )
X
gQ(xlx)
d~m(~)_]
0
n
x
=
exists
~
1=1
n
dY i
~
1=1
dF(x 1 ., ••• ,x.)
1
p1
f
X€R(€ )
x
=
ds (0)
m-
f£
f
X€R(€ )
~
n
~
1=1
dF(x 1 ·,···,x.)
1
p1
!((X'X + ; I)-l X'I - (X'X + I 1)-1 X'X 9
m
-
Jt •
X
n
+ (XiX +! I)-lxtxO_O JIM (XiX +! I)-lxiXO } 7d~ (0) ~ dF(x " ••• ,x .)
11
m
- In
- n -- 1=1
p1.
f
= \
n
X€R(€ )
+ m
tr~(X'X +! 1)-1 X' X(X'X +! I)-1M 7 ~ dF(x1., ••• ,x .)
m
m
- 1=1
1.
p1.
J'
X€R(€ )
n
x
j
X€R(€ )
~
1=1
dF(x ., ••• ,x .)
1 1.
p1
tr~(X1X
v -1
+ -)
m
-
n
M / ~ dF(x 1 ., ••• ,x.)
- i=l
1.
pl
=P
(€)
m
say.
13
It follows from lemma 2 that
m ->ro
,I
=v
Pm(€)
lim
trL(X'X)-1~7
dF
X€R(€ )
• • For sufficiently large m,
Pm(€)
>
J
v
trL(XtX)-l
~7
dF -7'2
X€R(€ )
>
v
trL(X'X)-l M_7 -
E
€
• • For such large values of m,
But for all
~,
r(~, !*)
=v
E
trL-(xtx)-l M_7
=
Sup r(~, !*)
9
Now if (10) is true, then
1\
Sup r(£,!)
= v
9
for sufficiently large
~0C
for arbitrary!.
.It
Sup
G
€~o
Sup
Hence
G
€
if
r(~m,
!)
m, which is impossible.
Hence (9) holds"
,
Thus,
Sup
r(G, t*)
G€"fo
,Now
€
A
<
Since
tr["(XIX)-l M_7 -
E
r(G, !*)
=
r(G, t*)
<
v
<
Sup
r(G)!)
G €
1
E
trL(X'X)-l
Sup
- -G€1j
r(G, t)
for all
!.
r(G, !*) •
for all
t.
5. Remarks •
a)
Condition (ic) is not satisfied in general.
In fact, it can be easily
seen that for
p
=1
14
and for a normal distribution of
satisfied if and only if
n
~
most all samples of size 2
b)
xl' this condition is
4, though t* can be uniquely determined for al-
or more.
It is obvious that when condition (ic) is not satisfied, the least
squares estimator is inadmissible.
c)
for
Nothing is known about the uniqueness of
t*
as a minimax estimator
Q or even its admissibility when the conditions of theorem
2
are satis-
According to Steinis ~27 conjecture, ~* may be shown to be inadmissible.
fied o
d)
No simple way of verifying condition (ic) is known.
However, under the
following modification of the sampling scheme, this condition is always satisfied.
Suppose (y, xl' ••• 'x ) follows a (p+l)-variate distribution.
p
and fix a constant
.
c, however large.
Let us choose
We then call the independent observations
(Yi' xli,···,x pi )' i=l, ••• ,n ~ p+l, on (y,
xl' ••• 'xp ) having risk of order c
.
or less i f and only if II (X 'X) -1 II < c • Then our sampling scheme is as follows:
Sampling Shceme.
Choose and fix a positive constant
independent observations on (y, Xl,.oe,X p ).
risk of order
c
If the
co :Make
n
x-observations have
or less, stop sampling; if not, reject the observations
and repeat the procedure till a set of observations having risk of order
c
or less is obtained, which is called the set
up to a risk of order
For any
or
effective observations
c.
c, the effective observations up to a risk of order
c
can be con-
sidered to be observations on a process (yi, Xii, •• o'X~i)' i=l, ••• ,n
Yi'.o.,y~
given
xli, ••• ,X pi ' i=l, ••• ,n,
gression function of
given
y'
1
on
for which
are mutually independent, the re-
Xi' ••• 'X~ and the conditional variance of
y'1
xi'.'.'x~ are the same as those for (y, xl, ••• ,x )' while the marginal
distribution of
p
(xi," .,x~)
satisfies condition (ic).
Hence if (y, xl' ... ,Xp )
satisfy condition (ii), then (y', xi' ••• 'x~) also satisfies condition (ii), and
similarly for condition (iii) or (iv) or (v).
It can also be noticed that we
15
have never made use of the independence of (Xli, ••• ,X pi )' i=l, ••• ,n
in course
of our analysis; al~.that we required was the independence of (Yl' ••• 'Yn ) given
(xli' •• o,x i)' i=l,o •• ,n
p
(Yi,
and this property is preserved in the process
....
Xii' ••• 'X~i)' i=l, ••• ,n.
Thus we see that under
(ii) and (iii), the least squares estimator
tive observations up
unique estimator for
1*
to a risk of some order
~
conditions (ia), (ib),
obtained from a set of effecc
belongs to (; l'
and is the
having uniformly minimum risk among all members of
~1
obtained from the same set of observations. Also, under conditions (ia), (ib),
(1i), (iv) and (v), the above estimator is minimax for
Q
in the class of all
estimators obtained from the same set of observations.
Under this sampling
scheme, the sample size becomes a random variable with expectation greater than
n but since
(X'X)-l exists with probability 1 by virtue of condition (ia),
the increase in the expected sample size over
by taking
c
n
can be made arbitrarily small
sufficiently large. Also, since no observation on
y
is necessary
in order to decide about the effectiveness of a set of (y~, X1i, ••• ,Xpi),i=1,ft.~,
and in many practical situations, an observation on (xl, ••• ,Xp )
costly than the observation on the associated
of observations due
is much less
y, the real increase in the cost
to the introduction of the above modification in the
sampling scheme may be much less than what appears at first sight.
The author wishes to thank Professor Wassily Hoeffding and Professor S. N.
Roy for making some helpfUl suggestions.
REFERENCES
~!7 Lehmann, Eo L o and Hodges, J. L e , Jr.(1950). Ann. Math. Stat., Vol. 2l,p.182.
l'g7 Plackett,
R. L. (1950).
~27 Stein, Charles' (1956).
Vol. I, p. 197.
Biometrika, Vol 36, p. 458.
Proc. Third Berko SymP. on Math. Stat. and Prob.,