Second Order Correctness of Perturbation Bootstrap M-Estimator of Multiple Linear Regression Parameter

摘要: Consider the multiple linear regression model $y_{i} = \boldsymbol{x}'_{i} \boldsymbol{\beta} + \epsilon_{i}$, where $\epsilon_i$'s are independent and identically distributed random variables, $\mathbf{x}_i$'s known design vectors $\boldsymbol{\beta}$ is $p \times 1$ vector of parameters. An effective way approximating distribution M-estimator $\boldsymbol{\bar{\beta}}_n$, after proper centering scaling, Perturbation Bootstrap Method. In this current work, second order results non-naive bootstrap method have been investigated. Second correctness important for reducing approximation error uniformly to $o(n^{-1/2})$ get better inferences. We show that classical studentized version bootstrapped estimator fails be correct. introduce an innovative modification in statistic modified pivot correct (S.O.C.) M-estimator. Additionally, we continues S.O.C. when errors independent, but may not distributed. These findings establish perturbation as a significant improvement over asymptotic normality M-estimation.