A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model

Statistical Papers, Jun 2010

Let \({\widehat{\varvec{\kappa}}}\) and \({\widehat{\varvec{\kappa}}_r}\) denote the best linear unbiased estimators of a given vector of parametric functions \({\varvec{\kappa} = \varvec{K\beta}}\) in the general linear models \({{\mathcal M} = \{\varvec{y},\, \varvec{X\varvec{\beta}},\, \sigma^2\varvec{V}\}}\) and \({{\mathcal M}_r = \{\varvec{y},\, \varvec{X}\varvec{\beta} \mid \varvec{R} \varvec{\beta} = \varvec{r},\, \sigma^2\varvec{V}\}}\), respectively. A bound for the Euclidean distance between \({\widehat{\varvec{\kappa}}}\) and \({\widehat{\varvec{\kappa}}_r}\) 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 \({{\mathcal M}}\) and sub-restricted model \({{\mathcal M}_r^*}\) containing an essential part of the restrictions \({\varvec{R}\varvec{\beta} = \varvec{r}}\) with respect to estimating \({\varvec{\kappa}}\).

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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)


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Pawel R. Pordzik. A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model, Statistical Papers, 2010, pp. 299-304, Volume 53, Issue 2, DOI: 10.1007/s00362-010-0336-3