Revised December 1989. Abstract: The universal optimality of some block designs with unequal block sizes is studied. under the usual homoscedastic model.
Statistics & Probability North-Holland
February
Letters 11 (1991) 177-180
1991
Universal optimality of block designs with unequal block sizes V.K. Gupta Indian Agrtcultural
Stotrstrcs Research Institute,
New Delht IIOO12, India
A. Das Indian Statistical
Instrtute,
Calcutta 700035, India
Institute,
New Delhi 110016, India
A. Dey Indian Stattstrcal
Received August 1989 Revised December 1989
Abstract: The universal optimality and under a certain heteroscedastic
Keywords:
Generalized
binary
of some block designs model.
balanced
block designs,
with unequal
heteroscedastic
1. Introduction
Kiefer (1958) introduced Balanced Block Designs (BBD) as a generalization of Balanced Incomplete Block (BIB) designs and proved the A-, D- and E-optimality of BBD’s in g(u, b, k), where g( v, b, k) is the class of all connected block designs with u treatments, b blocks, and constant block size k. Subsequently, Kiefer (1975) proved the universal optimality of BBD’s in g(u, 6, k). (For a definition of universal optimality, see Kiefer (1973.) Although a considerable amount of work is available on optimality of designs in g(u, b, k), not much appears to have been done on the optimality of designs with unequal block sizes, except for the recent papers by Lee and Jacroux (1987a,b,c), Dey and Das (1989) and Gupta and Singh (1989). The purpose of this paper is to study 0167-7152/91/$03.50
0 1991 - Elsevier Science Publishers
block sizes is studied.
under
the usual homoscedastic
model
model
the universal optimality of block designs with unequal block sizes. The results are first derived under the usual homoscedastic, fixed effects, model. Subsequently, a model in which the intrablock variance is proportional to block sizes is considered and universally optimal designs are derived under this heteroscedastic model. In proving universal optimality, use is made of a sufficient condition of Kiefer (1975, Proposition 1).
2. Preliminaries In the usual setting of block designs, let u be the number of treatments, b the number of blocks, and n the total number of experimental units. It is well known that under the usual homoscedastic, fixed effects additive model, the coefficient matrix of the reduced normal equations for estimating
B.V. (North-Holland)
177
Volume
11, Number
linear functions design d, is C, = R,-
2
STATISTICS
of treatment
& PROBABILITY
effects, using a block
NdKJ’N;,
(2.1)
where R,=diag(r,,, K,=
LETTERS
February
The following in the sequel.
well-known
diag(k,,,
k,,,...,
Lemma 2.4. For a binary block design d E 9( v, b, n), tr(Cd) = n - b, where tr( .) stands for the trace of a square matrix. q Lemma
k,,),
Definition 2.1. A design d E 9(0, b: k,, k,, . . ., k,,) is called a Generalized Binary Block (GBB) design if NJ has only two entries in its jth column, say xd, ( > 0) and yd, = xd, + 1, for j = 1, 2,. . . , b.
It is easy to see that xd, = [k/u] for j = 1, 2,. , 6, where [ .] is the greatest integer function.
2.5. For given positive integers s and minimum of Cl=,nf subject to I:=, n, = t, the n, are nonnegative integers, is obtuined t - s[t/s] of the n, are equal to [t/s] + s - t + s[t/s] are equal to [t/s]. 0
t, the where when 1 and
3. Universally
homo-
scedastic
Definition 2.2. A GBB design is called a Generalized Binary Balanced Block (GBBB) design if
optimal
We prove the following
%,,%I?,/k,
= x
fori#m,
i, m=l,2
X is a constant
Definition
independent
of i and m.
2.3. A design
nLl,,n,,,/kll,
tr(Cd)
k,,),
=n-
f: kl’
= A’,
where 178
i, m=l,2
h’ is a constant
,...,
2
n:,,.
I=1
tr(Cd*)
= 7:;
Definition
2.1 and
Lemma
completing
tr(Cd),
the proof.
Example
3.2. Let BIB design with k = 4, and A = 3, incidence matrix
0,
independent
N= of i and m.
0 N i
0
% be the incidence matrix of a parameters u = 9, b = 18, r = 8, and consider the design d * with N given by
/=I
fori#m,
theorem.
v,
d E 9( U, b, n) is called a Binary Balanced Block (BBB) design if nd,, = 0 or 1 and ;:
the
Proof. As per the sufficient condition of Kiefer (1975) a design d ~9 (the class of competing designs) is universally optimal in 9 if: (i) C, is completely symmetric, and (ii) tr(Cd) is maximum over 9. In the present case, complete symmetry of C,, is ensured by the definition of a GBBB design. Also, for any arbitrary design d E 9( U, b:
Using Cj’= ,ndr, = k,. 2.5, it follows that ,...,
under
Theorem 3.1. A GBBB design d* E~(u, b: k,, k,, . . . , kh), whenever existent, is universally optimal over 9( v, b: k,,k,, . , k,,).
J=l
/=I
designs
model
k,, k,,...,
where
will be used
rd2,...,rdl,),
rd, (k,,) is the replication (block size) of the i th treatment (jth block), and Nd = (n,,,) is the u X b incidence matrix of d. A design d is said to be connected if and only if Rank(C,) = LJ- 1. For given positive integers u, b, n, 9(u, b, n) will denote the class of all connected block designs with u treatments, b blocks, and n experimental units. Similarly, _@(u, b: k,, k,, . , kh) will denote the class of all connected block designs with o treatments, b blocks, and given block sizes k,, k,,. . ., k,. We may allow k, > u for some or all j = 1, 2,. . . , b.
f:
results
1991
21’ I,, + 11’ !
Volume
11. Number
2
STATISTICS
& PROBABILITY
where Z,,, is the m th order identity matrix, and 1 is a column vector of unities. Then d * is universally optimal in g(10, 27: k,, k, ,..., k,,), where k, = 4forj=l,2 ,..., 18and k,=12for18
