Stein’s Estimator versus Pre-Test 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 preliminary-test estimator, or a pre­test estimator for short. Sclove, Morris, and Radhakrishnan (1972) proved that Ida is dominated by Stein’s positive-rule estimator [ 1 — (doS/a’a)]+a for some do such that d<d0< 2(K — 2)/(n + 2).

A pre-test 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 = XA-1, 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

Подпись: (2.2.17)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

0

0

/«/

WJy*

WJy*

+ (1

-Q

J-

*

1____

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 positive-rule estima­tor у defined by (2.2.18) dominates the pre-test 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 Stein-type estimator over a pre-test estimator (with respect to a particular risk function), any Stein-type or ridge-type 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 = X-1, 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

(I-B)G4Q’4-c) 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’fi-C)«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

yJ-QGBG'(Q7?-c) C fi for fi. (2.2.26)

Let us consider a concrete example using generalized ridge estimator 1. Putting

Подпись: В =(K-2)a2^ a’ 22 a

Подпись: fi image160

in the left-hand 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 pre-test 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.

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