Unified theory of least squares

Radhakrishna Rao, C. (1973) Unified theory of least squares Communications in Statistics, 1 (1). pp. 1-8. ISSN 0361-0918

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Official URL: http://www.tandfonline.com/doi/abs/10.1080/0361092...

Related URL: http://dx.doi.org/10.1080/03610927208827002


Let (Y, Xβ, σ2I) where E(Y)=Xβ and D(Y) = E(Y→Xβ)'=σ2G, be the Gauss-Markoff model, where A' denotes the transpose of the matrix A. Further let β^ be astationary point (supposed to exist for all Y) of Y - Xβ)' M(Y-Xβ); i.e., where its derivative with respect to β is the zero vector. It is shown that if β^; is the BLUE of p'β for every P∈S(X'), the linear space generated by the columns of X', and an unbiased estimator of σ2 is ƒ-1(Y-Xβ^)' M(Y-Xβ^) f=R(G:X)-R(X), where R(V) denotes the rank of V, then it is necessary and sufficient that M is a symmetric g-inverse of (G+X∪X') where U is any summarice matrix such that S(G:X) = S(G + X∪X'). The method is valid whether G is singular or not and R(X) is full or not. A simple choice of U is always U=k2I, K¬0.

Item Type:Article
Source:Copyright of this article belongs to Taylor and Francis Group.
Keywords:Least Squares Theory; G-inverse; Linear Estimation; Gauss-markoff Model; Singular Dispersion Matrix; Unified Theory for Linear Models; Linear Hypotheses
ID Code:71688
Deposited On:28 Nov 2011 04:04
Last Modified:28 Nov 2011 04:04

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