污染数据线性回归模型的参数估计

Parameter Estimations in Linear Regression Analysis for Contaminated Data

污染数据线性回归模型在医学、现代分子学、气象预报学等多领域有着广泛的应用。笔者研究了污染数据线性回归模型:yi=α+βxi+iε,i=1,2,…,n,iε相互独立并且服从N(1μ,1σ2),同时{yi}受到另一串与之独立的随机变量{ti}的干扰,仅能观察到yi*=(1-v)yi+vti,0≤v≤1,其中ti(i=1,2,…,n)相互独立并且服从N(0,2σ2),1μ,1σ2,2σ2均已知。最后给出了回归参数α、β和污染参数v的估计。

The linear regression model of contaminated data is applied in many fields, such as medicine, science of modern molecular and meteorology. In this paper, we study the following linear regression model of contaminated data： yi=α＋βxi＋εi, i = 1, 2,…, n, where εi is independent and εi-N（μ1, σ1^2）. In addition, {yi} is disturbed by a series of random variables {ti} which are independent of yi. We now know yi^＊ =（1-v）yi＋vti, 0≤v≤1 , where ti（i =1, 2,…, n）is independent and ti-N（0, σ2^2）. Moreover, μ1 , σ1^2 , σ2^2 are given. At last, we give the estimation of regression parameters α,β and contaminated parameter v.