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Stahel–Donoho 稳健多元估计量的表现AI 解读

The Stahel-Donoho estimators (t, V) of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights of the form w(r), where w is a weight function and r is a measure of “outlyingness,” obtained by considering all univariate projections of the data. It has a high breakdown point for all dimensions and order √n consistency. The asymptotic bias of V for point mass contamination for suitable weight functions is compared with that…

基于投影的偏差稳健回归估计AI 解读

A new class of bias-robust estimates of multiple regression is introduced. If $y$ and $x$ are two real random variables, let $T(y, x)$ be a univariate robust estimate of regression of $y$ on $x$ through the origin. The regression estimate $\mathbf{T}(y, \mathbf{x})$ of a random variable $y$ on a random vector $\mathbf{x} = (x_1,\cdots, x_p)'$ is defined as the vector $\mathbf{t} \in \mathfrak{R}^p$ which minimizes $\sup_{\|\mathbf{\lambda}\| = 1} \mid T(y - \mathbf{t'x, \lambda' x}) \mid s(\mathbf{\lambda'x})$…

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线性模型中 M 估计量的渐近行为AI 解读

This paper deals with $M$-estimators for the linear model $y_i = \mathbf{x}'_i\mathbf{\theta} + u_i1 \leqslant i \leqslant n$, where the $\mathbf{x}_i$ are fixed $p$-dimensional vectors, and the $u_i$ are i.i.d. random variables with distribution $F$. The estimators considered are solutions $\hat\mathbf{\theta}$ of the equation $\sum^n_{j = 1}\psi(y_j - \mathbf{x}'_i\hat{\mathbf{\theta}})\mathbf{x_j = 0}$ for some function $\psi$. Let $\mathbf{X}$ be the matrix whose $i$th…

无限方差自回归过程最小二乘估计的渐近行为AI 解读

Let $y_t$ be an order $p$ autoregressive process of the form $y_t + \sum^p_{s=1} \beta_s y_{t-s} = u_t$, where the $u_t$'s are i.i.d. variables with a symmetric distribution $F$ such that $E \log^+ |u_t| < \infty$. For the Yule-Walker version $\beta_T^\ast$ of the least-squares estimate of $\beta = (\beta_1,\cdots, \beta_p)$, it is shown that $T^\frac{1}{2}(\beta_T^\ast - \beta)$ is bounded in probability.