Stein’s Estimator versus PreTest Estimators
Let a — N(a, cr2I) and S— <t2x„ (independent of a). Consider the strategy: Test the hypothesis a = 0 by the Ftest and estimate о by 0 if the hypothesis is accepted (that is, if S~la’a = d for an appropriate d) and estimate a by a if the hypothesis is rejected. This procedure amounts to estimating a by the estimator Idot, where Id is the indicator function such that Id — 1 if S~ la’a > d and 0 otherwise. Such an estimator is called a preliminarytest estimator, or a pretest estimator for short. Sclove, Morris, and Radhakrishnan (1972) proved that Ida is dominated by Stein’s positiverule estimator [ 1 — (doS/a’a)]+a for some do such that d<d0< 2(K — 2)/(n + 2).
A pretest estimator is commonly used in the regression model. Often the linear hypothesis Q’jJ = c is tested, and the constrained least squares estimator (1.4.11) is used if the F statistic (1.5.12) is smaller than a prescribed value and the least squares estimator is used otherwise. An example of this was considered in Eq. (2.1.20). The result of Sclove, Morris, and Radhakrishnan can be extended to this regression situation in the following way.9
Let R be as defined in Section 1.4.2 and let A = (Q, R)’. Define Z = XA1, у* = у — Xf, and у = A(fi — f) where f is any vector satisfying c = Q’f. Then we can write Model 1 as
y* = Zy + u. (2.2.16)
Partition Z and у conformably as Z = (Z1, Z2), where Z, = XQ(Q’Q)1 and Zj = XR(R, R)1, and as y’ = (y, yQ, where 7i = Q'(fi~ f) and y2 = R’O? — f). Then the hypothesis Q’j? = c in the model у = X0 + u is equivalent to the hypothesis y, = 0 in the model y* = Zy + u.
Define W = (W0,W,,W2) as W2 = (Z^Z,)1^; W, = (ZJZ,)"’^, where Z, = [I — Z2(Z2Z2)~1Z’2]Z1; and W0 is a matrix satisfying WqW, = 0,
W£W2 = 0, and W£W0 = I. Then we have
W’y* – A(W’Zy, <r2l),
and the hypothesis y, = 0 in model (2.2.16) is equivalent^) the hypothesis W’,Zy = 0 in model (2.2.17). Define Id = 1 if 5^ V %(ZjZ,) ‘Z’.y * > dand 0 otherwise and define 2) = {1 — [rf05’/y’*/Z1(Z,1Z1)1Z,1y’,‘]}+. Then the aforementioned result of Sclove et al. implies that
0 
0 

D 
W’y* 
1 ‘w’ + 
0 
w;y* 
W2y* 
dominates

in the estimation of W’Zy. Therefore, premultiplying by W (which is the inverse of W’), we see that
DZIZ’Zr’Z’y* + (1 – £>)Z2(Z;Z2)‘Z^y* = Zy (2.2.18)
dominates
/^(Z’Z^Z’y* + (1 – Id)Z2(Z&2)‘Z’#* m Zy (2.2.19)
in the estimation of Zy. Finally, we conclude that Stein’s positiverule estimator у defined by (2.2.18) dominates the pretest estimator у defined by (2.2.19) in the sense that E(y — y)’Z’Z(y — y) S E(y — y)’Z’Z(y — y) for all y.
Although the preceding conclusion is the only known result that shows the dominance of a Steintype estimator over a pretest estimator (with respect to a particular risk function), any Steintype or ridgetype estimator presented in the previous subsections can be modified in such a way that “shrinking” or “pulling” is done toward linear constraints Q’fi = c.
We can assume Q’Q = I without loss of generality because if Q’Q Ф I, we can define Q* = Q(Q’Q)1/2 and c* = (Q’Q)“1/2c so that Q*’fi = c* and Q*’Q* = I. Denoting the least squares estimator of fi by Д we have
Q’fi — c — N[Q’fi – c, <72Q'(X’X),Q]. (2.2.20)
Defining the matrix G such that G’G = I and G’Q'(X’X)1QG = X1, X diagonal, we have
G'(Q’fi – c) – N[G'(Q’fi ~ с), a2 X"1]. (2.2.21)
Therefore, if В is the diagonal matrix with the ith diagonal element Bt defined
under any of the minimax estimators presented in Section 2.2.6, we have
(IB)G4Q’4c) G'(Q’fi~c) for G'(Q’fi—c), (2.2.22)
where “0! < §2 for 0” means that
£(0, – 0)'(0i 0)^ E(62 – 0)'(02 – 0)
for all 0 with strict inequality holding for at least one value of 0. Therefore we have
GDG'(Q’fiC)«Q’ —c for Q’fi – c, (2.2.23)
where D = I — B, and consequently
^Q*c) + c]^Q fcr (2,,4)
where R is as defined in Section 1.4.2 with the added condition R’R = I. Then which can be simplified as
yJQGBG'(Q7?c) C fi for fi. (2.2.26)
Let us consider a concrete example using generalized ridge estimator 1. Putting
(K2)a2^ a’ 22 a
in the lefthand side of (2.2.26), we obtain the estimator
Aigner and Judge (1977) applied the estimator (2.2.27) and another estimator attributed to Bock (1975) to the international trade model of Baldwin (1971) and compared these estimates to Baldwin’s estimates, which may be regarded as pretest estimates because Baldwin utilized a certain linear restriction. Aigner and Judge concluded that the conditions under which Bock’s estimator is minimax are not satisfied by the trade data and that although Berger’s estimator (2.2.27) is always minimax, it gives results very close to the least squares in the trade model.
Leave a reply