The major contributions of this paper lie in two aspects. Firstly, we focus on deriving Bernstein-type inequalities for both geometric and algebraic irregularly-spaced NED random fields, which contain time series as special case. Furthermore, by introducing the idea of "effective dimension" to the index set of random field, our results reflect that the sharpness of inequalities are only associated with this "effective dimension". Up to the best of our knowledge, our paper may be the first one reflecting this phenomenon. Hence, the first contribution of this paper can be more or less regarded as an update of the pioneering work from \citeA{xu2018sieve}. Additionally, as a corollary of our first contribution, a Bernstein-type inequality for geometric irregularly-spaced $\alpha$-mixing random fields is also obtained. The second aspect of our contributions is that, based on the inequalities mentioned above, we show the $L_{\infty}$ convergence rate of the many interesting kernel-based nonparametric estimators. To do this, two deviation inequalities for the supreme of empirical process are derived under NED and $\alpha$-mixing conditions respectively. Then, for irregularly-spaced NED random fields, we prove the attainability of optimal rate for local linear estimator of nonparametric regression, which refreshes another pioneering work on this topic, \citeA{jenish2012nonparametric}. Subsequently, we analyze the uniform convergence rate of uni-modal regression under the same NED conditions as well. Furthermore, by following the guide of \citeA{rigollet2009optimal}, we also prove that the kernel-based plug-in density level set estimator could be optimal up to a logarithm factor. Meanwhile, when the data is collected from $\alpha$-mixing random fields, we also derive the uniform convergence rate of a simple local polynomial density estimator
翻译:本文的主要贡献在于两个方面。 首先, 我们的焦点是得出伯恩斯坦式的不平等类型, 包括几何直流和代数间间不规则的 NED 随机字段, 其中含有时间序列的特殊案例。 此外, 我们的结果通过将“ 有效维度” 的概念引入随机字段的指数集, 表明不平等的锐度仅与这个“ 有效维度” 相关。 根据我们所知的最佳数据, 我们的文件可能是第一个反映这一现象的。 因此, 本文的第一个贡献可以或多或少被视为来自\ citeA{xu2018sieve 的开创性工作的更新。 此外, 作为我们第一次贡献的必然结果, 伯恩斯坦式的不平等模式, 用于不定期空间 $\ alpha- 混合字段。 我们的贡献的第二个方面是, 基于上述不平等, 我们展示了基于内核的内核正调不正正正正正正正调的平流率值, 我们的平直径直径直的正正正正的平面的平面进程条件, 也可以在随后的平坦的轨道上, 。