Consider the partitioned linear regression model and its four reduced linear models, where y is an n × 1 observable random vector with E(y) = Xβ and dispersion matrix Var(y) = σ2 V, where σ2 is an unknown pos...Consider the partitioned linear regression model and its four reduced linear models, where y is an n × 1 observable random vector with E(y) = Xβ and dispersion matrix Var(y) = σ2 V, where σ2 is an unknown positive scalar, V is an n × n known symmetric nonnegative definite matrix, X = (X 1 : X 2) is an n×(p+q) known design matrix with rank(X) = r ≤ (p+q), and β = (β′ 1: β′2 )′ with β1 and β2 being p×1 and q×1 vectors of unknown parameters, respectively. In this article the formulae for the differences between the best linear unbiased estimators of M 2 X 1β1under the model and its best linear unbiased estimators under the reduced linear models of are given, where M 2 = I -X 2 X 2 + . Furthermore, the necessary and sufficient conditions for the equalities between the best linear unbiased estimators of M 2 X 1β1 under the model and those under its reduced linear models are established. Lastly, we also study the connections between the model and its linear transformation model.展开更多
For a singular linear model A = （y, Xβ, σ2 V） and its transformed model MF = （Fy, FXβ, σ 2FVF＇）, where V is nonnegative definite and X can be rank-deficient, the expressions for the differences of the estimat...For a singular linear model A = （y, Xβ, σ2 V） and its transformed model MF = （Fy, FXβ, σ 2FVF＇）, where V is nonnegative definite and X can be rank-deficient, the expressions for the differences of the estimates for the vector of FXβ and the variance factor σ2 are given. Moreover, the necessary and sufficient conditions for the equalities of the estimates for the vector of FXβ and the variance factor σ2 are also established. In the meantime, works in Baksalary and Kala （1981） are strengthened and consequences in Puntanen and Nurhonen （1992）, and Puntanen （1996） are extended.展开更多
基金supported by the National Natural Science Foundation of ChinaTian Yuan Special Foundation (No.10226024)Postdoctoral Foundation of China and Lab.of Math.for Nonlinear Sciences at Fudan Universitysupported in part by The International Organizing Committee and The Local Organizing Committee at the University of Tampere for this Workshopsupported in part by an NSF grant of China
文摘Consider the partitioned linear regression model and its four reduced linear models, where y is an n × 1 observable random vector with E(y) = Xβ and dispersion matrix Var(y) = σ2 V, where σ2 is an unknown positive scalar, V is an n × n known symmetric nonnegative definite matrix, X = (X 1 : X 2) is an n×(p+q) known design matrix with rank(X) = r ≤ (p+q), and β = (β′ 1: β′2 )′ with β1 and β2 being p×1 and q×1 vectors of unknown parameters, respectively. In this article the formulae for the differences between the best linear unbiased estimators of M 2 X 1β1under the model and its best linear unbiased estimators under the reduced linear models of are given, where M 2 = I -X 2 X 2 + . Furthermore, the necessary and sufficient conditions for the equalities between the best linear unbiased estimators of M 2 X 1β1 under the model and those under its reduced linear models are established. Lastly, we also study the connections between the model and its linear transformation model.
基金The project was supported by the Mathematical Tian Yuan Youth Foundation of China (10226024)Postdoctoral Science Foundation of Chinathe Science Foundation for Yong Teachers of Northeast Normal University.
文摘For a singular linear model A = （y, Xβ, σ2 V） and its transformed model MF = （Fy, FXβ, σ 2FVF＇）, where V is nonnegative definite and X can be rank-deficient, the expressions for the differences of the estimates for the vector of FXβ and the variance factor σ2 are given. Moreover, the necessary and sufficient conditions for the equalities of the estimates for the vector of FXβ and the variance factor σ2 are also established. In the meantime, works in Baksalary and Kala （1981） are strengthened and consequences in Puntanen and Nurhonen （1992）, and Puntanen （1996） are extended.
