The Related Efficiency Of The Least Squares Estimator In Singular Model

THE RELATIVE EFFICIENCY OF THE LEAST SQUARESESTIMATOR IN SINGULAR LINNEAR MODELWang JieMajor: Foundarnental Mathematics Graduate: Grade 97Advicor: Prof.Wang WeixiProf.Wang ChengrningIn recent years some statistiⁿs pay more attention to the relative efficiencies of parameter in Linear model. There are two important estimators: one is said the best Linear unbiased estimator. the other is the lease square estimator.are frequently used in estimation class of parameter. Because of some reason .we always use LSE instead of BLUE. This will result in loss to precision of estimator. So relative efficiency is introduced to measure this kind of loss. About this problem. some authors have discussed in Gauss-Markov model. In this paper, we discuss the lower bound of the relative efficiency in Singular Linear model. at the same time, the relation between relative effciency and gerneralized coefficiency of correlation are also built.We consider the Linear modelYzX13±c. E(^)O. Cov(^)a2E(lijwhere Y is an n x 1 observable random vector X is a design matrix. r(X) ,' < p.13 C R^. and a2 are unknown parameter. ï¿¡ is the n x 1 random error vector. E is a dispersion matrix r(⁾ = 2 n.On the basis of the theory of lease square, we know BLUE sf13.(X'T^XyX'T-Y,where A is Moore-Penrose general inverse matrix T E * X'UX where U Id. )c > 0 in thispaper, r(T) = r(^:X). ifs covariance matrix is Cov(j37) = (XfT^X)4X^TT*X(XcT^X)^.While the lease square unbiased estimator of 13 is - (X^X)±XIY . It's covariance matrixis Cov(13) = (X'X)+X'EX(X'X)+ a2 , According to Gauss-Markov theory.we knowCovCi3) Cov(8)In more cases of application .^l is unknow or not so clear, so lease square estmator ³ is used to replace ^. which will lead to some loss. An important way to estimate such loss is relative efficiency. Now. we-introduce some relative efficiencies as following:ei(/3) - flCov(/3^)Cov(13)fl4.X1(Cov(13j)(j3) - ^CovC3)IIFhere hAil denotes ^e-24 the general determinant of A ).1(A) denotes the largest characteristic root of A and Euclid norm of A respectively.Our main results are as follows:Theorem 3.1 Under the model (1). if p^X) C p⁾ thenei(rS) [iiⁿt.1 2 +here ) )' ) > 0 is the nonzero characteristic root ofT m> 2r.Theorem3.2 Under the model (1). if X is a column full-rank matrix and is nonsingular(1) - 'e³> J7fmatrix, n²p. then(2) e1QId)where ) A > 0 is the characteristic root of ⁿ(³.fl) is general coefflcientn of cOrrela^tionTheorem 4.1 Under the model (1),-' ²p&²A²_<sub><sub><sub><sub><sub><sub><sub>e2G3) ^Theorem 4.2 Under the model (1). suppose X is a column full-rank matrix and is nonsigular matrix .thene2(⁾ rc4^3./3here ^Y/.1r is a kind of generalized coef^cient of correlation defined by professor zhang yao ting.Theorem 5.1 Under the model (1),r^-<sup>-2⁴.,¹/2e3(fl) ^r-t+1 1m-r±iPr r-t^-2²y2Theorem 5.2 Under the model (1). suppose X is a column full-rank matrix and is nonsigular matrix, thenTheorem 6.1 el(I3)z1z^e3(I3)=1=^<sup>e2(I3>1.Theorem 6.2 et((3) e3(/3). equalicy is satisfied if and only if A - B.5Theorem 6.3 e1(/3) e2Qt3). equalicy is satisfied if and only if A = B. Theorem 6.4 Under the model (1). Assume?-ⁱ[²I^tthereforee2CB) e⁾equalicy is satisfied if and only if A B.Theorem 6.5 Under the mode] (1). for arbitaray two unbiased estimator of j3^. ifCovcBi) > Cov(/32) > 0there followse < e2) i 1.2.3which suggests that. the larger the biased error is. the smaller the relative eciency of estimator will be.</sup>