多元准正态线性模型中一个估计的非负最优性

AN OPTIMAL NONNEGATIVE PROPERTY OF AN ESTIMATOR IN MULTIVARIATE QUASI-NORMAL LINEAR MODELS

摘要: Y=X₁BX′₂+U_ε是一个多元线性模型,其中X₁,X₂和U≠0是已知矩阵,B是未知参数阵,ε是随机矩阵。假设ε有如下的一阶、二阶、四阶矩 \begingatheredE \varepsilon=0, E \varepsilon \varepsilon^\prime=1 \otimes \Sigma, \\ \operatornameCov \boldsymbol\varepsilon \varepsilon^\prime=2(I \otimes \Sigma) \otimes(I \otimes \Sigma)\endgathered其中∑≥0是未知参数阵.设∑^*是∑的最小二乘估计,C≠0是已知的非负定阵,本文对UU’是幂等阵的情形给出了tr（C∑^*）是tr（C∑）的最优非负二次无偏估计的充要条件。

Abstract: Let Y=X₁BX'₂+U_ε be a multivariate linear model, where X₁,X₂ and U≠0 are known matrices, B is an unknown matrix and ε is a random matrix. Suppose moments of the first, second and fourth order of ε have the forms as follows \begingatheredE \varepsilon=0, E \varepsilon \varepsilon^\prime=I \otimes \Sigma, \\ C_o v \varepsilon \varepsilon^\prime=2(I \otimes \Sigma) \otimes(I \otimes \Sigma)\endgathered where ∑≥0 is an unknown matrix and UU’ is an idempotent matrix. This paper gives a necessary and sufficient condition for tr （C∑）to be the uniformly minimum variance nonnegative quadratic unbiased estimator of tr （C∑） where ∑ is the least square estimator of ∑ and C≠0 is a known nonnegative definite matrix.