A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model
Pawel R. Pordzik
Let and r denote the best linear unbiased estimators of a given vector of parametric functions = K in the general linear models M = {y, X, 2V } and Mr = {y, X | R = r, 2V }, respectively. A bound for the Euclidean distance between and r is expressed by the spectral distance between the dispersion matrices of the two estimators, and the difference between sums of squared errors evaluated in the model M and sub-restricted model Mr containing an essential part of the restrictions R = r with respect to estimating .
1 Introduction and preliminaries
in which y is an n 1 observable random vector with expectation E (y) = X and
dispersion matrix D(y) = 2V , where the matrices X and V are known, both allowed
to be rank-deficient, while the vector and the positive scalar 2 are unknown
parameters. The model M is assumed to be consistent, that is, y C(X : V ). Let = K
denote a given vector of parametric functions linearly estimable in the model M, i.e.
C(K ) C(X ), and let stand for its best linear unbiased estimator (blue) in M.
Furthermore, consider the restricted linear model
obtained by supplementing the model M with linear constraints specified by an m p
known matrix R and an m 1 known vector r such that r C(R). Let r denote the
blue of in the model Mr .
The aim of this note is to constrain the Euclidean distance between the estimators
and r in terms that allow a clear statistical interpretation. The bound involves two
factors; the first one is the spectral distance between the dispersion matrices of the two
estimators (measuring the sub-optimality of the lue in Mr , or conversely, the gain in
matrix risk of the biased estimator r in the model M). The second factor, depending
on y through goodness of fit statistics, is the difference between sums of squared errors
evaluated in the model M and sub-restricted model {y, X | AR = Ar, 2V } with
implied restrictions being an essential part of R = r with respect to estimating ;
cf. Baksalary and Pordzik (1992).
Considering a linear model with nuisance parameters, the bound established in
Sect. 2 allows to assess how sensitive the estimation of the main parameters might
be with respect to possible overparametrization of the inference base. In this context,
some improvement of the result by Baksalary (1984, Theorem 2.4) is presented in
Sect. 3. A bound for the Euclidean distance between competing estimators is a natural
tool to explore geodetic data. For numerical examples, concerned with the precise
levelling problem, see Mkinen (2002) and Mkinen (2000); see also Schaffrin and
Grafarend (1986) for application to Global Positioning System (GPS) data.
2 Results and
Referring to the corner-stones of the inverse-partitioned-matrix method for statistical
inference in the general linear model M, assume that G1, G3 and G4 are any matrices
such that
X G1(X : V ) = 0,
V G1X = 0, (V V G1V )QX = 0, XG3(X : V QX ) = (X : 0)
XG4X = V V QX (QX V QX )QX V ,
i.e., the partitioned matrix ((G1 : G3) : (G3 : G4) ) is a g-inverse of the bordered
matrix ((V : X) : (X : 0) ); cf. Rao (1971, 1972). Then the blue of , its dispersion
matrix and the sum of squared errors in the model M can be expressed as = KG3y,
D( ) = 2KG4K and S S E = y G1y.
Let R1 = r1 be an estimable part of the restrictions R = r in the model
M, that is, R1 is a matrix such that C(R1) = C(R ) C(X ) and r1 = R1Rr.
Note that R1 = r1 can be written as the implied restrictions LR = Lr, where
L = I R0R0 with R0 = R(I XX); for the proof, observe that, by the
equality C(R ) C(X ) = C(R (RX )), we have R1 = LR. Baksalary and Pordzik
(1989, Theorem 1) represented the consistency condition, the blue of and the sum
of squared errors for the restricted model Mr in terms referring to the model M and
a subset of estimable restrictions R1 = r1. The results useful for our purposes are
given in the following lemma.
Lemma 1 The restricted model Mr = {y, X | R = r, 2V } is consistent if and
only if the model M is consistent, i.e. y C(X : V ), and
1 r1 C(S),
where S = R1G4R1 while 1 = R1G3y is the blue of 1 = R1 in M. If the model
Mr is consistent, then the blue of and its dispersion matrix are
r = CS( 1 r1) and D(r ) = D( ) 2CSC ,
where C = KG4R1 and is the blue of in the model M. Moreover, if S S Er and
S S E are the sums of squared errors in the models Mr and M, respectively, then
S S Er = S S E + ( 1 r1) S( 1 r1).
Recall that, as far as only the estimation of in the model Mr is concerned, some
further reduction of the initial constraints is possible. Namely, R1 = r1 can be
reduced to a subset of implied restrictions which states the so-called essential part of
R = r with respect to estimating = K. Concerning the problem of reducing the
linear constraints in the restricted model Mr , Baksalary and Pordzik (1992, Theorem
2) showed that the blue of in the sub-restricted model {y, X | AR = Ar, 2V }
continues to be the blue of in Mr if and only if C(C ) C(SB ), w (...truncated)