The Superiority About A Class Of Linear Estimation Of Regression Coefficient Under Pitman Closeness Criterion And Best Affine Unbiased Response Decomposition

This thesis consists of two research papers concerning some special fields of probability theory and mathematical statistics. It is designed into two chapters.Chapter One consists of one paper which is related to the superiority about a class of linear estimation of regression coefficient under Pitman Closeness Criterion. Let the growth curve model be Y_n×p = A_n×mB_m×kC_k×p + E_n×p,E ～N(0,σ²I_n(?)I_p). Suppose that the least squares(LS) solution and linear estimation of regression coefficient are (?) = (A^TA)^-A^TYC^T(CC^T)^-1 and (?)₁ = (A^T A +ρ∑)^-1 A^T YC^T (CC^T)^-1 , when A A is ill-conditioned, where p is a positive constant, ∑ is a positive definite matrix. On the condition of exchangeability or unexchangeability of A^TA and ∑,we prove that under suitable conditions the linear estimator (?)₁ is better than B by Pitman Closeness criterion, and we extend the above result when A^TA and ∑ are both ill-conditioned.Chapter Two consists of one paper which is related to best affine unbiased response decomposition. This chapter contains two parts.Given two linear regression models y1 = X₁β₁+u₁ and y2= X₂β₂+u₂, where the response vectors y1 and y2 are unobservable but the sum y = F₁y₁ + F₂y₂ is observable,where F is nonsingular.We study the problem of decomposing y into components (?)₁ and (?)₂, intended to be close to y1 and y2,respectively.The first part:a necessary and sufficient condition for the existence of an linear affine unbiased decomposition,(?)₁ and (?)₂, is given. Under this condition, we establish the existence and uniqueness of the linear best affine unbiased decomposition and provide an expression for it.The second part supposed F₁ = F₂ = I,a necessary and sufficient condition for the existence of an nonlinear affine unbiased decomposition,(?)₁ and(?)₂,is given. A sufficient condition for the best nonlinear affine unbiased decomposition is also given. Under these conditions,we establish the existence and uniqueness of the nonlinear best affine unbiased decomposition and provide an expression for it.