Minimal sufficient statistics for incomplete block designs with interaction under an Eisenhart model III

Journal of Statistical Planning and Inference 19 (1988)317-324

Department of Statistics,

317

Dallas, TX 75275, U.S.A.

North Teas State University,Denton, TX 76203, U.S.A.

East TennesseeState WniveFsity,Johnson City, TN 37614, U.S.A. Received 9 January 1986; revised manuscript received 7 August 1987 Recommended by W.J. Studden

Abstract: The purpose of this paper is to derive minimal suffi complete block design and the group divisible partialiy bala ode1 III (mixed model) is assumed. The the and Hirotsu’s (1965)for the same model wit Gra dition of a statistic, Cii I$. A

Subject Classification: rimary 62510; Second

10.

Key words: inimal sufficient statistics; balanced incomplete bloc design; group divisible partially balanced incomplete block design.

318

C.H. Kapadia et al. / Minimal sufficient statisticsfor IBD

eeks and Graybill, 1962). Sufficient statistics are not only useful in identifying minimum variance unbiased estimates, they are also someti gesting prior distributions of the appropriate parameters in nderson (1980) develop ). Stroup, (AConja, squares computing algorithm for maximum Ii elihood estimat ean rather than all ponents using the sufficient statistics and a function of t observations, as the other procedures require an erived a minimal set of sufficient statistics

otsu (1965) who showed similar result for s fication model under to some special designs in&ding tistics for each case and he also gave a e model considered by these authors included no block and treatment interaction component but, in practice, we usually e an interaction component and hence eeks (1963, 1984) conered the two-way classification model with interaction un 1 derived a minimal set of sufficient statistics for the t paper is concerned with the derivation of minimal sufficient statistic; and G with interaction under a usable form of the statistics explicitly. The results are compared to the cases when there are no interaction components.

n this section we will be concerned with the balanced incomplete block design with a component of variance due to block-treatment interaction. Under an eeks (1963), the matrix ode! for this

ere ‘IT is a vector of f c

s,

C.H. Kapadia et a/. /

nimal sufficient statistics for

319

(2.2)

(2.3) where al = 1h2,

~%ch can bc ~rified by showing that CC- ’ = L To determine a set of sufficient statistics we examine the exponent, QI of the likelihood function,

fter expanding (2.7) and using (2.61, we find that =a1

- 2azt

+ (terms not including (here J= Jr). consider those terms in (2.5) that are

C.H. ~Q~adi~ et al. / ~i~i~a~ suf~cient statistics for iBD

320

Letting S/ =Si wkre

is replaced by

+a2f3+a3f4~2

a2~ifSi

s i=

J, 2, . ..A

then

~~.~~

z

re fi = Sl-- SF, f2i = S2i- S$, etc. S2 m3d S5 we see that there is a redundancy in these two statistics, since t i=t

Szr =

i.e.

a

om (2.7), f 3

C.H. Kapadia e? al. / Minimal sufficient statisticsfor IBD Then

(2.10)

321

ca

= t,

results obtaine odell with no in

we can deter-

CA

322

Kapadia et al. / Minimal sufficient statistics for IBD

tals can take the place of t independent linear combinaticns

of the

(obviously cs t). To prove minimality of the sufficient statistics in

proceeding as in the f’s are defined as in the previous section. sufficient statistics S,‘i- Si” and S&Ois are minimal in the orb=c,aseto procedure used for the hen for b> c,

I

is assumed in a G

with interaction

b> c there are c t - l), c linearly indepen imal sufficient set ofstatistics if b > c and there are b + t + 11 t-n), b)) statistics in a minimal

otals) statistics in a

Y,i*

(j=l,*a**

set if b=c. These results are identical to the those obtained by irotsu (1965) for the s e model without inter of the statistic 0 X$.

ltquist and Craybill(l965) n, except for the addition

o show that the w’s (as efined by (2.11)) form a linearly ‘=W,

hen

independent

w29 w3, w49 W2l,***,WZ.I9 w51,**.,W5I)*

where

and * 1 0 0 0.

;!

.

0

0 - 1 1 0

1 0 - 1 0

0 0 0 1

0 0 0 1

.. . ... .. . .. .

0 0 0 0

0 0 0 0

... ... ... ...

0 0 0 0 . ’

0 0

0 0

1 0

0 ... - 1 ...

1 0

0

P

0 .. .

,.”

1 .. . 0

0

0

0

0 .. .

I

i 1I

-

ii

il

1 .

set, let

323

y row 0 0

1 0 0

1 0 -1 cs 0

0

0

0

1

0 :

0

0

0

0

0

0

0

1 0 0 0

0

-1

1 1

0 0 0 0

..* ... .. . .=.

..1

0

1 ...

0 0 0 0 0

0 .*.

1

0 0

0

1 .. . 0 :

0 . .. 0 0 ..* 0 0 0 0 *a.

l

.

Ii

-1

...

o...

-1

.. 0 :

..*il

=

[

0 *. ; l

and so 0 -I I 0

and so bank = 2t + 4. Now, A is of di Thus det(A*) = )c2?+3, then ran (A *) < 2t + 4, a c~~t~adic(2t + 4), so rank )r2?+3. Ifran tion. Thus ~~~(~) =2t+ 3, and so q, ?Vzi’S,I939 Wd ZUid pIv5i’S ~~~~~~ 4%~~~~~~~~~ illdependent szt.

Afonja, B. (1972). Minimal sufficient statistics for variance csmponcnts for a general class of designs.

~i~rnetrik~59, 295-302. Ferguson, T.S. (196. ’ ~Qt~ernutic~iStatistics - A Decision TheoreticApproach. Academic Press, New York. 1wQ~ ~ombi~ng inter-block and intra-block information ia balance Graybitl, F.A. and D.L. Weeks &__,_ -. incomplete blocks. Ann. Math. Statist. estimator for Hirotsu, C. (1965). Research for a set of m aftsufficient statistics an 1. Res. ~~. ~~p~~Sci. BIBD and GD-~~~~D. Rep. ~tfftist t statistics for the two-way C~~ssjf~cat~Q~ Huitquist, R.A. and Graybill, EA. (1 ). Variance components in two-way c~~ss~f~cat~~~

C.H. Kapadia et al. / Minimal sufficient statisticsfor 1BD

324

Scheffi

Anderson (1980).Maximum likelihood estimation o IB design, Comm. Stat&. Theory and Methods A Wet ts, D.L. and F.A. Oraybill (1962). A minimal sufficient statistic for a general class of designs, .,

J.W. Evans and

(1950). ChnpletaseSS, SiiiiilZtr regiWiS afib uribkiised estimation.