关于误差方差估计渐近正态收敛速度的上、下界

TWO-SIDED BOUNDS ON THE RATES OF CONVERGENCE TO THE NORMAL DISTRIBUTION OF THE ESTIMATION OF ERROR VARIANCES

摘要: 考虑线性回归模型 Y_i=x'_iβ+ρ_i, i=1,2,…设误差序列e_i,i≥1满足条件:e_i,i≥1 i.i.d.,E \epsilon_1=0, E \epsilon_1^3=\sigma^2>0, \infty> Var \theta_I^2=\tau^2>0,记\hat\sigma_n^2=\frac1n-r\left\\sum_i=1^n \mathcalA-\sum_k=1^r\left(\sum_i=1^n a_n k, p\right)^2\right\, 其中:\sum_j=1^n a_n t f a_n m y=\delta_l m \quad n=1,2, \cdots, \delta_l m是kronecker符号。本文证明了:存在常数\sigma_1, \sigma>0使得\beginaligned & \sup _\varnothing\left|P\left(\frac\hat\sigma_n^2-\sigma^2\sqrt\text Var \hat\sigma_n^2\sigma_1 \delta(n),\endaligned。

Abstract: Considering the linear regression modelY_i=x'_iβ+ρ_i, i=1,2,…. Suppose that e_i,i≥1 be i.i.d, random Variables with Ee₁=0, Ee₁²=σ²＞0, ∞＞Var e₁²=r²＞0. Let \hat\sigma_n^2=\frac1n-r\left\\sum_j=1^n e_j^2-\sum_k=1^r\left(\sum_j=1^n a_n k j e_j\right)^2\right\, \beginaligned& \left.\delta(n)=\tau^-2 E\left(\theta_1^2-\sigma^2\right)^2 I_\left(\theta_1^2-0^2\right.>\sqrtn \tau\right)\endaligned,where \sum_i=1^m a_n j j \omega_n m s=\delta_l m and \delta_i m is kronecker sign. In this paper, we prove that there exist real numbers \infty>C, \sigma_1>0 suth that \beginaligned & \sup _\varnothing\left|P\left(\frac\hat\sigma_n^2-\sigma^2\sqrt\text Var \hat\sigma_n^2\sigma_1 \delta(n),\endaligned.