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摘要翻译：
我们考虑了单个（非随机）序列$(x_1,y_1)，(x_2,y_2)，...\in\real\times\real$的单变量回归估计，它在以下意义上是稳定的：对于每个区间$a\子集q\real$,(i)在$x_1,x_2,...$下的$a$的极限相对频率由未知概率分布$\mu$决定，(ii)在a$中的$y_i$与$x_i\的极限平均由未知回归函数$m(\cdot)$决定。给出了一个简单的估计$m(\cdot)$的方案，并证明了对于稳定序列$\{(x_i，y_i)\}$是$l_2$一致的，使得$\{y_i\}$是有界的，并且$m(\cdot)$在$(-i,i]$,$i\geq1$上的变化存在一个已知的上界。证明了对于回归函数具有有限变化的稳定序列族，即使在[0,1]$和$y_i$是二元值的限制下，也不存在一致的估计方案。
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英文标题：
《Regression estimation from an individual stable sequence》
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作者：
Gusztav Morvai, Sanjeev R. Kulkarni, Andrew B. Nobel
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Computer Science 计算机科学
二级分类：Information Theory 信息论
分类描述：Covers theoretical and experimental aspects of information theory and coding. Includes material in ACM Subject Class E.4 and intersects with H.1.1.
涵盖信息论和编码的理论和实验方面。包括ACM学科类E.4中的材料，并与H.1.1有交集。
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一级分类：Mathematics 数学
二级分类：Information Theory 信息论
分类描述：math.IT is an alias for cs.IT. Covers theoretical and experimental aspects of information theory and coding.
它是cs.it的别名。涵盖信息论和编码的理论和实验方面。
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一级分类：Mathematics 数学
二级分类：Statistics Theory 统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics 统计学
二级分类：Statistics Theory 统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
We consider univariate regression estimation from an individual (non-random) sequence $(x_1,y_1),(x_2,y_2), ... \in \real \times \real$, which is stable in the sense that for each interval $A \subseteq \real$, (i) the limiting relative frequency of $A$ under $x_1, x_2, ...$ is governed by an unknown probability distribution $\mu$, and (ii) the limiting average of those $y_i$ with $x_i \in A$ is governed by an unknown regression function $m(\cdot)$. A computationally simple scheme for estimating $m(\cdot)$ is exhibited, and is shown to be $L_2$ consistent for stable sequences $\{(x_i,y_i)\}$ such that $\{y_i\}$ is bounded and there is a known upper bound for the variation of $m(\cdot)$ on intervals of the form $(-i,i]$, $i \geq 1$. Complementing this positive result, it is shown that there is no consistent estimation scheme for the family of stable sequences whose regression functions have finite variation, even under the restriction that $x_i \in [0,1]$ and $y_i$ is binary-valued.
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PDF链接：
https://arxiv.org/pdf/710.2496
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