Bernstein-type Inequalities and Nonparametric Estimation under Near-Epoch Dependence

Abstract

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 $α$-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 $α$-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 $α$-mixing random fields, we also derive the uniform convergence rate of a simple local polynomial density estimator \cite{cattaneo2020simple}.

Figures (2)

• Figure 1: Green dots denote the selected locations. These locations are selected from the candidate set, which is the union of the $H_{0}$ blow-ups of lines A, B, and C, where $H_{0}$ is universally finite.
• Figure 2: The $N_k$s denote the sample size and satisfy $N_k\leq N_{k+1}$. Based on the definition of $R_{N_k}$ and $\Gamma_{N_k}$, we have $\Gamma_{N_k}\subset \Gamma_{N_{k+1}}$. Thus, Figure 2 illustrates the variation of $R_{N_k}$ as the sample size grows. Assume that the selected locations (green dots) lie only within a finite blow-up of the blue line. If the change in the blue line in some given direction is uniformly bounded, we can regard $\max_{k\in\mathbf{N}}H_{0k}$ as a finite positive number, since an infinitely long band with finite bandwidth is large enough to contain all locations (green dots) in $\Gamma_N$. Meanwhile, the change in the blue line cannot be a graph, such as the Archimedean spiral, which is therefore excluded.

Theorems & Definitions (19)

• Proposition 1
• Definition 1
• Proposition 2
• Definition 2
• Definition 3
• Definition 4
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6