论文标题
对指数族的回归函数的强大估计
Robust estimation of a regression function in exponential families
论文作者
论文摘要
我们观察到$ n $对独立（但不一定是i.i.d.）随机变量$ x_ {1} =（w_ {1}，y_ {1}），\ ldots，x_ {n} =（w_ {n}，w_ {n}，y_ _ {n}），并解决条件分布的问题，并解决条件分布的问题。 $ q_ {i}^{\ star}（w_ {i}）的$ y_ {i} $给定$ w_ {i} = w_ {i} = w_ {i} $ for ALL $ i \ in \ in \ in \ in \ in \ {1，\ ldots，\ ldots，n \} $。即使这些可能不是事实，我们的估计器也基于以下假设：$ y_ {i} $给定的$ y_ {i} $的条件分布$ w_ {i} = w_ {i} = w_ {i} $属于一个一个参数gropential gropential $ \ bar bar {\ mathscr {\ nathscr {q native parameter Space of parameter Space $ Intives $ Intives $ i $ i。更准确地说，我们假装这些条件分布采用$ q _ {{\boldsymbolθ}（w_ {i}）} \ in \ bar {\ bar {\ bar {\ mathscr {q}} $的某些$ {\boldsymbolθ} $属于vc-class $ \ barsym的$ i \ bolds $ i的$ i;对于每个$ i \ in \ {1，\ ldots，n \} $，我们估计$ q_ {i}^{\ star}（w_ {i}）$通过相同形式的分布，即\ bar {\ Mathscr {q}} $，其中$ \ hat {\boldsymbolθ} = \ hat {\boldsymbolθ}（x_ {1}，\ ldots，x_ {n}）$是$ \ bar {\ bar {\baldsymbolθ} $。我们表明，我们的估计策略对于模拟错误指定，污染和异常值的存在是可靠的。此外，当$ \ bar {\ bar {\BoldSymbolθ} $是低尺寸或中度维度的VC级功能时，我们提供了计算$ \ hat {\boldsymbolθ} $的算法，我们进行了模拟研究，以比较$ \ hat {\ boldsymbolth} $和基于Med的Med and Med和Med and Med and Med and Med and Med and Med and Med and Med and Med and Med and Med and Med and Med and Med。
We observe $n$ pairs of independent (but not necessarily i.i.d.) random variables $X_{1}=(W_{1},Y_{1}),\ldots,X_{n}=(W_{n},Y_{n})$ and tackle the problem of estimating the conditional distributions $Q_{i}^{\star}(w_{i})$ of $Y_{i}$ given $W_{i}=w_{i}$ for all $i\in\{1,\ldots,n\}$. Even though these might not be true, we base our estimator on the assumptions that the data are i.i.d.\ and the conditional distributions of $Y_{i}$ given $W_{i}=w_{i}$ belong to a one parameter exponential family $\bar{\mathscr{Q}}$ with parameter space given by an interval $I$. More precisely, we pretend that these conditional distributions take the form $Q_{{\boldsymbolθ}(w_{i})}\in \bar{\mathscr{Q}}$ for some ${\boldsymbolθ}$ that belongs to a VC-class $\bar{\boldsymbolΘ}$ of functions with values in $I$. For each $i\in\{1,\ldots,n\}$, we estimate $Q_{i}^{\star}(w_{i})$ by a distribution of the same form, i.e.\ $Q_{\hat{\boldsymbolθ}(w_{i})}\in \bar{\mathscr{Q}}$, where $\hat {\boldsymbolθ}=\hat {\boldsymbolθ}(X_{1},\ldots,X_{n})$ is a well-chosen estimator with values in $\bar{\boldsymbolΘ}$. We show that our estimation strategy is robust to model misspecification, contamination and the presence of outliers. Besides, we provide an algorithm for calculating $\hat{\boldsymbolθ}$ when $\bar{\boldsymbolΘ}$ is a VC-class of functions of low or moderate dimension and we carry out a simulation study to compare the performance of $\hat{\boldsymbolθ}$ to that of the MLE and median-based estimators.
