自变量带误差的线性模型参数估计的收敛速度

A CONVERGENCE RATE OF ESTIMATES OF PARAMETERS OF THE MULTIVARIATE “ERRORS IN VARIABLES” MODEL

给出了部分变量带误差而部分变量不带误差的线性回归模型的参数估计的收敛速度．

? We discuss the multivariate “errors in variables” model in this paper. The independent variables a 1t ,a 2t ,…,a pt , satisfy a linear regression model, i.e. ∑α ia it =μ. Some of these independent variables, such as a it (i=r+1,…,p), can be observed without errors in them, while the observations of other variables can be obtained with errors, i.e. χ it =a it +ε it (i=1,2…,r t=1,2,…,n) are the observations of a it (i=1,2,…,r t=1,2,…,n), and χ it =a it for i=r+1,…,p and t=1,2,…n. Under the conditions that ε t=(ε 1t ,ε 2t ,…,ε rt ) τ t=1,2,…,n are i.i.d. and E(ε i)=0, E(ε i·ε τ t)=σ 2·I, it is given the follow convergence rate of estimates of parameters: ξ,ζ,μ and σ 2: ||| -1 -|| -1 ζ|=0[(n -1 log log n) 1/2 ] a.s. |-ζ|=0[(n -1 log log n) 1/2 ] a.s. |-μ|=0[(n -1 log log n) 1/2 ] a.s. | 2-σ 2|=0[(n -1 log log n) 1/2 ] a.s. where ξ=(α 1,α 2,…,α r) τ, ζ=(α r+1 ,α r+2 ,…,α p) τ 