ESTIMATING THE MODEL VARIANCE OF A RANDOMIZATION-CONSISTENT
REGRESSION ESTIMATOR
Phillip S. Kott and K.R.W. Brewer
National Agricultural Statistics Service, Fairfax, VA 20230, USA and
Australian National University, ACT 0200, Austra lia
KEY WO RDS: Asymptotic; Calibration equation;
Weighted-residual-mean-squared-error estimator
1. INTRODUCTION
Suppose we want to estimate a population
total, T = 3U yk, based an a sample, S, of n elements.
Randomization-based theory tells us we can do that
with a regression estimator of the form:
t = 3 (yk /Bk)
k0S
(1)
+ [ 3 xk ! 3 (xk /Bk)]( 3 xk!xkdk /Bk)
-1 3 xk!ykdk /Bk ,
k0U k0S k0S k0S
where Bk is the sample selection probability of element
k, xk = (xk1 , ..., xkP) is a row vector of values associated
with element k, 3U xk is known, and the dk are arbitrary
non-negative constants. Särndal, Swensson, and
Wretman (1989) call t a “general regression estimator”
or GREG . From a mod el-based point of view,
however, t is not very general. That is why we do not
use that name here.
The estimator t can be written as t = 3S akyk,
where ak = (1 /Bk) + [ 3U xi ! 3S (xi /Bk)](3S xi!xid i /Bi)
-1
xk!dk /Bk. Often the dk in equation (1) are chosen so that
these ak have desirable properties (e.g., being positive;
see Brewer, 199 9). The ak have been constructed in
such a way that the calibration equation (Deville and
Särndal 1992), 3S akxk = 3U xk, is satisfied.
Under mild conditions, t is randomization
consistent (see Isaki and Fuller, 1982, who use the
synonymous term “design consistent;” B rewer, 1979,
introduces a similar property). We will not be deeply
interested in randomization-based properties here. Our
focus, instead will be on the properties of t as an
estimator for T under the linear model:
yk = xk$ + ,k,
where $ is an unspecified P-member column vector,
E(,k *zk) = 0 for all k 0 U, and zk = (xk, Bk, dk). It is
easy to see that t is a unbiased estimator for T under the
model in the sense that
)