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APPLIED MATHEMATICS AND STATISTICS LADORATORIES

STANFORD UNIVERSITYCALIFORNIA

GENERALIZED LEAST SQUARES ESTIMATORS FOR

RANDOMIZED FRACTIONAL REPLICATION DESIGNS

BYS. ZACKS

TECHNICAL REPORT NO. 86March 15, 1963

PREPARED UNDER CONTRACT Norw-225(52)(NR-34 2-022)

FOROFFICE OF NAVAL RESEARCH

(

11

IIRANDOMIZED FRACTIONAL REPLICATION DESIGNS

Ii byS. Zacks

TECHNICAL REPORT NO. 86

March 15, 1963

i I PREPARED UNDER CONTRACT Nonr-225(52)(M-342-022)II: ASTIA

IFOR 1-7r7,'nm' 7I OFFICE OF NAVAL ]RESEARCH APR 10 1963

J ~J L LaTISIA

Reproduction in Whole or in Part is Permitted forany Purpose of the United States Government

APPLIED MATHIEATICS AND STATISTICS LABORATORIES

STANFORD UNIVERSITYI

STAI•iORD, CALIFORNIAII

I

1 Generalized Least Squares Estimators

ifor Randomized Fractional Replication Designs

by

I S. Zacks

1. Introduction

Fractional replication designs have become of great importance,

I especially for industrial experimentation. A missile whose operation

is affected simultaneously by dozens of interacting factors would pro-

duce a full factorial experiment of impractical size. Indeed, if there

are more than 20 factors which may affect the operation of a missile,

and if we like to attain complete information on all the main effects

and interactions of the controllable factors we would run the experi-

20ments over more than 2 = 1,048,576 treatment combinations. Fractional

replication designs are planned to attain information about some of

the main effects and interactions by a relatively small number of trials.

If the operation of a missle can be controlled with some information on

I the main effects and some low order linear interations, it might be

I sufficient to run only 32 or 64 trials at a time. These however, should

be chosen from those possible in some optimal manner.

f The problem of choosing a 1/ 2 m-s fractional replication and an

appropriate estimator of the parameters characterizing the factorial

model (main effects and interactions) has been studied by A. P. Dempster

(1960, 1961), K. Takeuchi (1961), S. Fhrenfeld and S. Zacks (1961,

1962) , S. Zacks (1962), B. V. Shah and 0. Kempthorne (1962a,b). In

all these studies the type of estimators considered is that which yields,

under a randomized procedure with equal probabilities of choice, unbiased

I

estimates of a specified linear functional of a subvector of parameters,

which lies in the range of the design matrix (the matrix of the corres-

ponding normal equations).

In the present study statistical properties of the generalized

least squares estimators, under randomized fractional replication de-

signs, are studied. The term generalized least-squares estimators (de-

noted henceforth by g.l.s.e.) is used since the matrices of the nor-

mal equations corresponding to these designs are singular. The fac-

torial models corresponding to the type of fractional replication de-

signs studied in the present paper is presented in section 2. For this

sake we start from the factorial model for a full factorial system.

Then we present the required algebra, and the method of constructing

the orthogonal fractional replications. The linear spaces of all g.l.s.e.

associated with the various orthogonal fractional replication designs

are characterized in terms of the linear coefficients of the corres-

ponding factorial models. Some statistical properties of the g.l.s.e.

under procedures of choosing a fractional replication at random, are

then studied. First we prove that there is no g.l.s.e. which yields

unbiased estimates of the entire vector of in parameters. However,

there are g.l.s.e.s which estimate unbiasedly subvectors of parameters.

The trace of the mean-square-error matrix corresponding to a g.l.s.e.

applied under certain randomization procedure is used as a loss

function for the decision problem of choosing a g.l.s.e. and a ran-

domization procedure. It is shown that when the parameters of the

factorial system may assume arbitrary values, the randomization pro-

cedure which assigns equal probabilities to various fractional

2

I

replications (denoted by R. P.*) is admissible. Bayes g.l.s.e., relative

to a-priori information available on the parameters, are then studied.

This leads to a minimax theorem, which specifies a minimax and admis-

sible g.l.s.e, under R.P.o*

The relationship between the generalized inverse of the matrix of

normal equations and g.l.s.e. as given by A. Ben-Israel and J. Wersen

(1962), and by C. R. Rao (1962) is studied. It is shown that these

are particular cases in the general class of g.l.s.e. studied presently.

Finally, it should be remarked that although the present paper

deals with factorial system of order 2m all the important results

hold in more general factorial systems of order pm (p > 2).

3

2. The statistical model for fractional replication designs.

2.a. The statistical model for a full factorial experiment

of order 2 m

A full factorial experiment of order 2m is a set of 2m treat-

ment combinations, consisting of m factors X0 ,..., X each at

two levels. Such a system can be characterized by 2m parameters

ao,..., , which are the coefficients of the multilinear re-2 m1l

gression function:

(2.1) E(Y(Xo,...,Xm.i ) X 0 u(, Xo Xl.. " m" 1

where X 0,1 (j=0,...,m-l); Uxo~,...,x 1 ) 2J;S 0 ----

and Y(Xo,...,X m_1) is a random variable representing the "yield" of

the experiment at treatment combination (X ,...,X M_). Denote by

X ,0 and X ,l (j=O,...,m-l), XJ, 0 < Xj,I, the two specified levels

of factor X . By changing variables according to the transformation

Xj~ i (Xj~ + XI(2.2) ZJ,k 1Jk I (x- 2 Jil (k=0,l;j--O,...,m-l)

f (xil XJ, 0 )

the regression function (2.1) is reduced to the form:

(2.3) E(Y(Zo,.. z )) = Pu Z 0 Z1 " m-Io _ 1m u--O u M-

where Z =1-, +1 (J=O,...,m-l); and u=U(o,...,X )

j 'm-4

I

Writing Zj = (-1) J with i j= 0,1 for all J = 0,...,m-1 , the

regression function (2.3) can be represented in the form:

M-12l M_ 1 E % (1-i )

(2.4) E(Y(i 0 ,..,i m-1 = ou(-l ) J=Ou=O

rm-1Furthermore, denote by xv (1.i M-ir ), v I . 2J , the

J=O2 m treatment combinations of the factorial system under considera-

tion; and let

m-1

(2.5) c( 2m) (-l)j=O , for all v,u = 0,..., 2 m 1vu/

then (2.4) is reduced to the form:

2m-1 (2m) m(2.5) E[Y(x)) = c ,u for all v = O,

Let Y' = (Y(X 0 ),...,Y(x .)) be the vector of observations at all

the 2m treatment combinations; and let 1' = ( PO 2m-l) be the

vector of parameters of (2.5). Thus, if

(C() = cm) , (vu = ,...,2 M),

denotes the matrix of the coefficients of the P's in (2,5), then

the statistical model for a full factorial system can be written as:

(2.6) Y = (C (2m)) P + C

where E is a random vector, with E c = 0 and E CE' = a2 1 ( 2m)

(In) denoting the identity matrix of order n).

5

!

2.b. The algebra of factorial experiments.

2.b.l. Properties of the matrices (C(2m))

The properties of the matrices (C (2m)) will be presented without

proofs. For details see S. Ehrenfeld and S. Zacks (1961).

The matrices (C (2m)) , m = 1,2,..., of (2.6) can be obtained

recursively by the following relationship:

(2.7) (C() = -1f (c(2) ,- m = 1,2,....

(where (C(0)) 1 (scalar), and where A®B is the Kronecker's

direct multiplication of the matrix A by the matrix B from the

left, defined as follows: If A is an n x m matrix ilaiji1, and

B is a k x 1 matrix lb rsi, then A(B is the nk x ml matrix:

• I o

anl B a 'a BI " I a*

I I

From the associative property of the Kronecker's direct multiplica-

tion it follows that every matrix (C(2m)) can be factorized into

2m'sx 2m-s (1 < s < m) submatrices of order 2s x 2s , according to

the relationship:

(2.8) (C(2m)) = (C(2m-s)) (C(2)) , (1 < s < M)

6

1From this relationship it follows that the elements of (C (2m)) are

related to those of (C ( 2 m-s) and (C(2s)) according to:

(2.9) C (2m)o =C (2 ) c(2m's)(.)i+J2It irt Jqt

for all i = 0,...,2 -1; j = 0,...,2 -1; and where t = rt + t2r

(r t = 0,...2S8-l ; qt = Op'"".2m-'sl) .

Another useful relationship among the elements of (C ( 2 m)) is

the following one: Let ulu 2 = 0,...,2 M- be given by

Ui (%oi, X i,..., I(ml)i) (i=1,2); 0j,1 = O,1 (J=O,...,m-1), and

define u u (o' Q' where Xm X + X (mod. 2);1 2 o M- 1 J 1 .Jl 2

then

(2.10) C(2m) = ( 2 m) ' (2m) for every v=O,...,2m-iVU1 U2 v1u1 v'u 2

The properties of (C(2)) are extended into (C(2m)) by the

recursion relationship (2.7), and are summarized as follows:

(i) C(2) = 1 for every v =v,0

(ii) C(2m) 1 for every u = 0,...,2m-12m-l,u

01 (2m)m(iii) I C = 0 for every v = 0,...,2 -2u=O v

u011• (2m)_m(iv) I C - 0 for every u = l,...,2m-i for every

v=O m

U = 1,...,2-1.

!

2(-) 7 (2m) c(2m) 2M)ifU(v) E~ fv if vv=O l u2 0 , if u 1 u 2

and

(2m) f2m if v1 'v2( 2T) 7 M Vlu vM -

(vi) u--O 2 0, if v 1 v2

(v) and (vi) can be expressed also in the form:

(C(2m)) (c(2)) - (c(2m))(c(2m)) = 2m(2)

2.b.2. The group of parameters.

Every parameter Pu (u = 0 ,.o., 2 m-1 ) of the statistical model

(2.6) can be represented by an m-tuple Pu = (ko' ,''X M-l) where

k = 0,U (J = O,.,m-1)

The set of all 2m parameters constitutes a group, B with

respect to the operator Q, defined as follows:

m-1 m-1Let u X k' 2j and u 2 = • X•21 then k

j=0 j=O 1 2

if, and only if k = 1 where V X + X' (mod. 2) forJ=O

all j = 0,...,m-l.

The unit element of the group B is P - (0,0,...,0) and the

1

inverse of M- (Xo. ., i.) is - (2-Xo,2.XlI 2-% l)(mod. 2)."" -l oU )•* , -, iu. ,

A set of n parameters uI )u 2'...'"u is called dependent if there2 n

I

exist n constants a k (k-l,...,n), not all of which are zero

(ak = 0,1), such that:

(2.11) [u11 a 2 2 [Un = poU1 a12 U 0

In

O , if a = 1where whr , if a = 0

If relationship (2.11) is valid only when all ak = 0 (k=l,...,n)

then Ul'''"Un are called independent. It is easy to check that

every set of n independent parameters (1 < n < m) generates a sub-

group of order 2 in B.

2.c. The construction of a 1 / 2 m-s (1 < s , m) fractional

replication.

Let ( '0dod 1 Pdm-s1 ) be any set of m-s independent

parameters in B. The 2 m treatment combinations can be classified

into 2m's disjoint subsets Xv (v=O,...,2m-s-l) of equal size, re-

lative to the specified m-s independent parameters, in the following

manner: Let Pdk E (X od kIXdk,..., XMlid k = 0,...,m-s-l be

a defining parameter, and let X a (io,...,i M.) be a treatment com-

bination, then x E X if, and only if,

m-1(2.12)O j 'j, ak (mod. 2), and where

-s-i ak 2

k=O

I

In order to perform the classification of treatment combinations into

the blocks X (v = 0 ,..., 2 ms- 1 ) we do not have to solve the linear

equations (2.12), but it suffices to compare the rows of the matrix

of coefficients (C( 2 m)) under the columns corresponding to the

special independent parameters (0d '0''"d ). These two procedureso mn-s-i

are equivalent (see S. Khrenfeld and S. Zacks (1961)). The m-s inde-

pendent parameters, relative to which the classification takes place,

are called defining parameters. The answer to the question, which of

the parameters should be specified for the role of defining ones de-

pends on the objectives of the experiment. The choice of a set of de-

fining parameters will generally effect the bias of estimators and

their variances, and might have other effects on the properties of

statistics and procedures (see 0. Kempthorne (1952); S. Ehrenfeld and

S. Zacks (1961).

The term fractional replication, in its broadest sense, relates

to any subset of treatment combination from a full factorial system.

K. Talnkuchi (1961) considers designs of randomly combined fractional

replications. We shall consider in the present paper only fractional

replications which consist of one block of treatment combinations, Xv,

chosen from the set of 2ms blocks constructed according to the pro-

cedure outlined above. These fractional replications are called

orthogonal. A randomized fractional replication procedure is one in

which a block Xv is chosen with a probability vector ý' = ( .toA 2m-s 1 )

2.d. The statistical iodol for a 1/ 2 m's fractional replication.

Let P 'd 0* d*']3d ) be a set of defining parameters;2m r-s I

(Xv ; v = 0,..., 2Ms- 1) the corresponding blocks of treatment combinations,

10

I

and Y(Xv) the vector of observations associated with the 2s treatment

combinations in Xv. The order of the components of Y(Xv) is deter-

mined by the standard order of the corresponding x's in Xv, e.g. if

Xv = (Xo, x5, xs, x6 ) then Y(Xv)' = (Y(xo), Y(x3), Y(x5 ), Y(x6 )).

Let P(0)= (Po' 1. t2sI) be any specified vector of 2s

parameters independent of the defining parameters (except for the "mean"

Po); with t < t for all k = l,...,2 S-1. Let (1* ; u=O,...,2m'-i)o k k+1 u

be the subgroup of 2m-s parameters, generated by the m-s defining

parameters. Define by P (u) (u = 1,...,2m's-l) the vector of 2s

parameters obtained by multiplying each of the components of 0(0) by

P, i.e., 0(u) = (Po(D , P t ® 3),... t2l@ ) . Then, the

statistical model for Y(Xv) can be written in the form:

2m-s.-

(2.13) Y(X) = v (p( 2S)) P + E = (p )P* + Cu=O

where; as proven by S. Ehrenfeld and S. Zacks (1961)

=( 2 -s " " b ( 2 M -s ) ) ( : O- 2 s )(2.lh) 1 (i, bvl ,.M_ v( 2m!l)

is a 2 sx( 2m- 2 s) matrix; (P(2)) is a 2s x 2s matrix obtainedvO

from (C(2m)), by picking the elements of (C(2m)) corresponding to

treatment combinations in X and the parameters in P(0) andV

arranging them in the standard order. The scalars b(2), by which-•(2s)p(2s)vu

we multiply (P 2)) to obtain (P (2) are given by the formula:vo vum-s-i

( 2 . 5 )( ) ( i 5 ( i j - L ( d j ) )

(2.15) b( u (-1)J:°

11

Irn-B-i mn-s-i m r-i j

where v = 1 ij2J;u= u iý 2j and L(dj) X ,d (mod. 2),

J=o Jýo k=O '

for every defining parameter and where.

and E is a random vector of order 2s, with Ec = 0 and EEE' a c2 1(2s).

(p_(2s))It can be readily proved that, the rows of ( vu ) (v'u--O'''"2')'

as well as its columns, are orthogonal, i.e.,

(2.16) (P(u) (P(2) = u(2s))((2s)) = 26 2

for all v,u=O,...,2m-l; and that a similar property holds for the

matrix (Bd "m))' whose elements are the coefficients b (2m-s) u,dS.. .,d S-)

defined by (2.15), i.e., j

(2.17) (B do,.? .,dm-s)l))t (B(do,...,dns~ i)) = 2m-s I( 2 mrs)

for every choice of m-s defining parameters. For the sake of simpli-

fying notation, let S - 2s and M = 2m-s, N = S-M = 2 m. Furthermore,

assume, without loss of generality, that the defining parameters are thek(o ) ' s l

"main effects" (PSP2S ,."'N/2) and that P = ()•, P,.

then, the blocks of treatment combinations are:

(2.18) (Xi+vS ; i = 0,...,S-1) for all v = 0,..., M-I

and the statistical model for Y(Xv) is given by:

(2.19) Y(Xv) = [(i, C(M) C(M) )D (C (S) ]Pvvi "'" v(M-1)(D((S)

(c(S))0(0) + M-1 0(M) P(u) + EU=l

where P (u) =(pus' Pl+us''" P(u+l)s-i)'

12

3. Generalized least squares estimators for fractional rplications.

t 3.a. The set of all least squares estimators.

jGiven a block of treatment combinations X (v=O,...,M-1) and the

associated vector of observations Y(Xv), the "normal equations" cor-

responding to the linear model (2.19) are given by:

(3.1) (C_)', tCv) = (Cv)' Y(XV) I v=O,...,M-I

where (Cv) is the S x N matrix of the coefficients of (2.19), i.e.,

(5.2) (C ) = t (, ) c(M) ) ( s)v vk '"'" v(M-l)

A generalized least squares estimator (g.l.s.e.) of P is any

linear operator (Lv), on E(S) (Eucliden S-space), so that (Lv)

is an N x S matrix satisfying the equation:

(353) (C V)' (Cv)(Lv) = (Cv )' , v=O,...,M-l

Let (Lv)' - ((Lyo)' (Lv)' (L(Ml))') where (L)

(u=O,...,M-l) are square matrices of order S x S. Substituting

from (3.2) for (Cv) in 3.3 and decomposing (Lv) as indicated

we arrive at the matrix equation:

(Lvo) 1

(3.4) s[(Q(M) )®(D(IS) ".. ........ (S)),

(L ) C(M)Lv(M-1) Lv(m-1)

where (Q(M)) (l,C(M.) C(M) ), (lC(M) C(M) is aS 'vl'" v(M-1) vi "'' v(M-I)

13

square symmetric matrix of order M x M, whose (i,J)-th element is

qM(M) M) C(M) (iJ=O,...,M-l). Since C(M) = + 1 and (C(S)qij = vi vj vu --

is non-singular, the linear equations in the matrices (L vu) can be

expressed in a form equivalent to (5.4) as,

M-( (M) (c(S)) I(s)

u=O

Since the unique solution to the equation (H)(C(S)) I I(s) is

(H) =A (C (S) it follows that the M matrices ( v ), J=0,...,S

M-l, whose submatrices are given by:

1 c(M) (c(S)), if u = j-l(3.6) S vuIvu (0) , otherwise

constitute a basis of M independent solutions of (3.3). Thus,

every g.l.s.e. (L ) can be represented as a linear combination of

the M linearly independent operators (L(J) ) (J=O,...,M-l), so

that the coefficients of (LvJ)) add up to 1. Formally, the set of

all g.l.s.e., given X is:v

M-1 Lj -(.7) •(C) = (L") : (Lv) = Ml M-j ) =

1 =0 j =o

Every g.l.s.e. can thus be represented by M coordinatesM-1 A

(Xo, 1%,...,X . 1 ) such that 2 Xj = 1. Furthermore, if P v de-J=O

notes the vector of g.l.s.e. of P then we have, according

to (5.6) and (3.7)

14

!

X 0 C(M)

(3.8) v 1 c vu (Q (c (SY)) Y(Xv) for every v=O,...,M-l.C (M-1)SM-1 vu

S3.b. Som__.e statistic-al properties of glse

In the present section we prove that there are no unbiased g.l.s.e.

of P, and derive an expression for the trace of the mean-square-error

Amatrix of a g.l.s.e. P.

Consider a fractional replication design in which a block Xv

(v=O,...,M-l) is chosen with probability tv (tv > 0 for allM-1

v=O,...,M-l; 2 tv = 1). A randomized fractional replication pro-v=O

cedure is thus represented by an M-dimensional probability vector t.

This class of randomization procedures contains, in particular, the

fixed fractional replication design, in which one of the X blocksv

is chosen with probability one.

Theorem 1.

Let (u) = C(M) (C(S))' Y(X) then E (•(u)) P •(u) for

all u=O,...,M-l if, and only if, E* = , 1

Proof.

The expected value of ý(u) under randomization procedure t

is given by:

15

I

(3-9) E• (P (u)) 1 t •v C S Y(X)

' M- 1 (M) (S) Lw c(M) (c(S))P(w)I tv C'U (C) IL + (c)(

-1 M-1 cC (M) C(M) = •(u) + M1 M-1

w=O v=O vu w0 (w,)= O t v

• (M) C(M) (w)'vu! vV

Clearly, if tv = 1 for all v=O,...,M-l, thenMM-I

M-1 c(M) (M) i 1 c(M) c(M)

v=O v=O

for all u / w by the orthogonality of the column vectors of (C(M)).

Thus, E. ((u)) = (u) for all u=O,...,M-l. On the other hand,

if Et (•(u)) = 1(u) for all u--O,...,M-I then, in particular

(3.10) E() P() M-l M-1 '(M) P (w)(3.zo)E•{•(O I I t(o ÷ v c

w=l v=0O

Bu te onitonM-l •(M)B e1 C = 0 for all w=l,...,M-I is equivalent

v= VW

to the condition:

Multiplying both sides of (3.11) by (C(M)) we get:

16

I

(3.12) Mt = (c(M))(c(M))I t =(C(M))(:)= 1 (M)

where 1 (M) is an M-dimensional vector with unity in all its components.

It follows that a necessary condition for the unbiasedness of (u is

that (M) i.e., each block is chosen with the same probability.M

(Q.E.D.)

Returning to the g.l.s.e. we have:

(3.13) E {A = Et'(( 0 p (O)' X lP)))

1 X(M) X 1 ) '

where t* = 1 = ( ,M

M-1Since Z X = 1 we conclude that there is no unbiased g.l.s.e, of P.

u=O

The g.l.s.e. in which X 0 = 1 and X = 0 for all u > 0 yields0 u

unbiased estimates of the components of P(0) only. Similarly whenX= i (J:O,...,M-1) and Xj, = 0 for all j' i J, the corresponding

g.l.s.e. yields unbiased estimates of the components of ' (W only.

AThe mean-square-error dispersion matrix of a g.l.s.e. P , under

randomization procedure t, is defined by E . Let

M(f,X;P) denote the trace of the mean-square-error dispersion matrix

of a g.l.s.e. represented by a vector X, such that V'1 (M) = 1, under

randomization procedure • ; i.e., M(t,X;P) = EL((•-$)' (t- )P

17

Theorem 2. 1The trace of the mean-square-error dispersion matrix under ran-

domization procedure t is given by the expression:

2M112 X M-1 •(u) E2(3.14) M(t,X P) = ( + iZ) (2- u-I)l +

u=O u=OM-1 (-1 M-1 (M) (M) (u1)' (u2)

+uI =07 (2X u +1) (uuI__ L v-° CvuI C2 0

where i1i 2 = Pp, (u)12 = P(u),p(u) (u=O,...,M-1)

Proof. I

According to (3.8)

M-1 2(3.16) E• 03'P) 2 E _X u Y(Xv)

Substituting (2.19) for Y(Xv) in (3.16) we get:

1 1 M-1 (S) (ul)

(3.17) E (Y(X)' Y(x)) = is E[[ Z (C )PS t v u! =0 V

M-1 C ) (C(S)) P(u2)+ Eu2 =0 vu2

2 M-1 M-1 (u 1 (u2)2 + E( I C) c(M) (u()' ,C() ) (

ul=O u=0 vu 2

a 2 + M•1 p(u)12 + Mu1 t u (M) C(M) (u 1 (u2)

U-O u 1 /u2 v=O 1 2

18

!Furthermore, E = (X Et •(O)'P ()1 X E0M:)' )

Substituting (3.9) for Et ((u)), we arrive at

(1 M-12 M-1 M-(I) (

E3.18= P E iv C c()u=O U =10 u 2u 1

1 1 2

(Ul)' (u2)]

Thus, from (3.15)-(3.18) the result holds.

Q.E.D.

Corollary; When each block X (v--O,...,M-l) is chosen with equalv

probabilities (f=t*) we have

2M-1I M-1 i(u) 12(3.19) M(f*,'X;) = (2+ IP12) M-l - I (2Xu.1)

u=O u--O

3.c. Optimum strategies

A strategy of the Statistician is a pair of two M-dimensional

t vectors (f,X) such that t is a probability vector, and %' 1 (M) 1.

Every strategy (f,%) represents a randomization procedure and a

g.l.s.e. The decision problem is to choose (f,X) optimally, with

respect to the loss function M(f,X;P).

Comparing (3.14) to (3.19) it is easily verified that for every

(f,X) there exists fo in En su2h that M(t,X;P°) > M(t*,kX;°) .

Thus, whenever P is arbitrar, t* represents an admissible randomi-

zation procedure. For this reason we shall restrict the discussion

from now on to strategies with randomization procedure t*, and turn

now to the problem of deciding upon an optimum g.l.s.e. under •*.

19

We notice in (3.19) that M(t*,X;•) depends on P only through

the M values 1P(u) 12. An a-priori information concerning these Ivalues might thus be utilized for the choice of X. Thus, let 7(u)

be an a-priori distribution of 1P(u) 12 , defined over the half-line

[0,oa).

Theorem 5.

The Bayes g.l.s.e. of 0, with respect to the a-priori distribu-

tions [FT(O),...,TT(Ml)} , under randomization procedure V is

(M-1))determined by the vector X=, (X(O), ... I , , where

(320 (u) E 7(u)(5.20) iiu M-1 ( for all u=O,...,M-I j

u=0 (u)_I E MuOP 12

Proof:

The risk P'unction under (•*,X) and 7 is

(3.21) R(t*, X;TT) = ( 02 + M-1 E O P(u 12)) M _- 1 (2x2u -l)

I ý~u) Z :(2 1U=O E)Q1-u)2] u0 U=O

E (u)(0 •12)

It is easily verified that X(U) (u=O,...,M-l), given by (3.20),yrM-1

minimize (5.21) under the constraint I x = 1.u=O

Q.E.D.B•)(u)2 M-Il~ )te

Let R(u) = E (u)[C 12) (u=O,...,M-l) and R = thenyr u=O

the Bayes risk with respect to an a-priori distribution 7- is

20

2 M-1 (u2(3.22) R(t*,X (Mf) - (02 + R) it (x(u)) 2

-

U-0

- 1 (2x(U) R(u)- -( ' 2 M-. (U) 2

u=O ix it

In particular, when all 1P(u) 12 (u--O,...,M-l) have the same a-priori

distribution, with R(u) = R* for all u=O,...,M-l, then the Bayesi It

g.l.s.e. is represented by X* = ( 1 ., with a Bayes risk

2(3.23) R(-*,x*,T) =.-+ (M-1)R*

M i

Theorem 4.

* = 1 l represents the minimax and admissible g.l.s.e.M

under randomization procedure k* relative to the class of allM-I

a-priori distributions T , such that R = R (u)

2u=O(C < R i< ). The minimax risk is given by (5.23).

Proof:

The minimax risk is the maximal Bayes risk, with respect to all

the a-priori distributions 7 in the class considered. The BayesM-1 (u)

risk for any of these v's is given by (3.22) where R = uu=O

is a given constant. Set the Lagrangian

(3.24) L(R(o),...,R (M1) ;P) = H' L 2.. M-( (u) )2+

+ -M-1

R(u)uu=O

21

By differentiating partially with respect to R(uj (u=O,...,M-l) and

p and equating the derivatives to zero we arrive at the system of

linear equations:

2 i-2 R(U)(- 1 2 2 • ) = R for all u--O,...,M-l

(3.25) (--

-I R~u Ru=O

The solution of this system of linear equations is given by R(u) R

2 M

for every u=O,...,M-l. Furthermore, since R > a , all the second

order partial derivatives with respect to R.u) are negative. Thus all ia-priori distributions n such that R(u) =

v W - for every

u=O,...,M-I are minimax strategies for Nature. As mentioned before, I1 l1 (M)

* = lM is then the unique minimax strategy for the Statistician.Mi

The Bayes risk corresponding to X* is given by (3.23). The admissi-

bility of X*, relative to the class of a-priori distributions con-

sidered, follows from the fact that it is the unique minimax.

Q.E.D.

22

4. The genraized inverse and the g.l.s.e.

4.a. The g.l.s.e. of minimum norm

A. Ben-Israel and S. J. Wersan (1962) proved that the g.l.s.e.

(L v) with minimum norm, i.e., min tr (L v),(~ ,v is a particular

v ~(L) v

generalized inverse (C v)I of the matrix of coefficients in (2.19),

namely (C) (1,1 C M ...1 (M~) ) X (C (S) ). The generalized in-

verse (C v)t always exists, it is unique, and given in general by

the formula:

(4.1) (C ) = [I (N- (D )(D' D f(D )'( vvC

for all v=O,...,M-l; where (E v) is a product of elementary trans-

formations, which transforms (C )'(C)v into:

(4.2) (E v)(X )'(X) ..[ . ..... (v=0 . . . M-l)

L(0) (0)-and where

(4.3) (D v) v0..M

The generalized inverse matrix, (C V) has the properties:

(4.4) (C v)(C v)t(C v) (C V) for all v=0,.,.,M-l

and

(C V) (C )(C )' P (C v)

A straightforward computation of (C v)t according to formula (4.1)

yields the result

23

"[(o) 1(4.5) (C, M" I

nM-d)

That is, (Cv)t is a g.l.s.e. represented by X* = (M) and hasM

the optimal properties mentioned in the previous section. This re-

sult can be obtained in the present framework more easily. According

to (3.8) a g.l.s.e. is given by

(M)

[M-i C(v(Ml1

Accordingly, the norm of (Lv) is

(4.7) tr. (Lv)'(L): tr. (L M-l x2 (C(S))(C(S))P)

S u--O

M-I M-I=Etr. I xU I I "

u=O u=O

Since the vector X* = l mi M-1 %2 under the constraintSine heveto k = minimizes I

M-1MZO x = 1 , it follows that the g.l.s.e. represented by X* minimizes

= U

the norm of (Lv), (v=O,...,M-l).

4.b. The g.l.s.e. suggested by C. R. Rao.

C. R. Rao (1962) defines the g.l.s.e. of P by the operator

(Lv) = [(Cv)'(C)v" (Cv)' where [(Cv)'(C )v] is a generalized in-

verse of (Cv)'(Cv). In case (C)' (Cv) is invertible, (Lv)" is the

24

!

unique g.l.s.e. of A. We shall prove now that under the present model

of fractional replication designs, Rao's g.l.s.e., (Lv)- , is represented

by the vector X" = (i,0,0,...,0). For this purpose, define the matrix

of elementary transformations

l 11c(M) o1 Vl . (s)(4.8) (E ) o ". (E S

_C(M) 1v(M-1)

then we have, for every v=O,...,M-l,

(4.9) (Ev)[(C)'(C)](E )' = (o) +1(0 (0)]

Hence,

(4.10) [Cv)'(Cv)] : (Ev)'(E)

To show this, consider the relationship

(4.11) [(Cv),(Cv)][(Cv),(c)]Cv) ,(cv)] = [(Cv)'(Cv)]

Multiply both sid"s of (4.11) from the left by (E) and from the

right by (Ev)'. Then,

(4.12) (Ev)(C)'(Cv)I(Ev)'(Ev )[(Cv)'(Cv)](E) =v

Or according to (4.9)

25

'I

I r rS0'(l (0)o

. [(o) (0o)] 1o0) (0 1 (0) (0) 1

Accordingly,

(4.14) (L) (Ev)'CEv)(Cv

M -CvM ... Cv(M.±) 1

_(M)1 "vl 0 '@ (s)S(M) . c(M)

•(M-) .0 • v(M-1) j"Ucv(M-l)

1 0 (C(S))' [~0]

That is, Pao's g.l.s.e. (L v is represented by %_ = (1,,...,0)'.

From theorem 1 it follows that (L )_ is an unbiased estimator

of (P(0),O). The trace of the dispersion matrix of (Lv)_ Y(XV)

under randomization procedure t* is given according to (3.19) by

(4.15) M(t*X;) = 02 + 2 Ila2 - 21P(O)12

2 M-1 P(u)12

u=l

Thus, in case all 1P(u) 12 have the same a-priori distribution, the

risk under strategies (l*,X) and n will be

26

1. (4.16) 2 0 + 2(M-1) R*

IComparing (4.16) to (3.23) we conclude that Rao's g.l.s.e. (Lv

1 might be very far from the optimum g.l.s.e. in case all the subvectors

of P have approximately the same average effect. On the other hand,

in case the effects of P(1) to 0(M-1) are negligible relative to

the effect of P(0) the Bayes g.l.s.e. will be very close to Rao's

g.l.s.e.

Acknowledgment

The author gratefully acknowledges Professors H. Solomon and

H. Chernoff for their helpful comments and encouragement.

27

References

1. A. P. Dempster (1960), "Random allocation designs I: On general tclasses of estimation methods", Ann. Math. Stat. Vol. 31,pp. 885-905.

2. A. P. Dempster (1961), "Random allocation designs II: Approximativetheory for simple random allocations", Ann. Math. Stat. Vol. 32,pp. 387-405.

3. S. Ehrenfeld and S. Zacks (1961), "Randomization and factorial ex-periments", Ann. Math. Stat. Vol. 32, pp. 270-297.

4. S. Ehrenfeld and S. Zacks (1962), "Optimum strategies in factorialexperiments", submitted for publication in the Ann. Math. Stat.

5. 0. Kempthorne, The Design and Analysis of Experiments, John Wiley,New York (1952).

6. C.R. Rao (1962), "A note on a generalized inverse", Jour. Roy.Stat. Soc. (B) Vol. 24, p. 152.

7. B.V. Shah and 0. Kempthoren (1962a), "Some properties of randomallocation designs", Technical Report, Stat. Lab. Iowa Stat. Univ.

8. B.V. Shah and 0. Kempthorne (1962b), "Randomization in fractionalfactorial", Technical Report, Stat. Lab. Iowa State Univ.

9. K. Takeuchi (1961), "On a special class of regression problems andits applications: Random Combined Fractional Factorial Designs",Rep. Stat. Appl. Res. JUSE, Vol. 7, No. 4, pp. 1-33.

10. K. Takeuchi (1961), "On a special class of regression problems andits application: some remarks about general models", Rep. Stat.Appl. Res. JUSE. Vol. 8, No. 1, pp. 7-17.

11. S. Zacks (1962), "On a complete class of linear unbiased estimatorsfor randomized factorial experiments", submitted for publication inthe Ann. Math. Stat.

28

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