Bernstein-type inequalities and nonparametric estimation under near-epoch dependence

The main contributions of this paper are twofold. First, we derive Bernstein-type inequalities for irregularly spaced data under near-epoch dependent (NED) conditions and deterministic domain-expanding-infill (DEI) asymptotics. By introducing the concept of “effective dimension” to describe the geometric structure of sampled locations, we illustrate – unlike previous research – that the sharpness of these inequalities is affected by this effective dimension. To our knowledge, ours is the first study to report this phenomenon and show Bernstein-type inequalities under deterministic DEI asymptotics. This work represents a direct generalization of the work of Xu and Lee (2018), thus marking an important contribution to the topic. As a corollary, we derive a Bernstein-type inequality for irregularly spaced $\alpha$-mixing random fields under DEI asymptotics. Our second contribution is to apply these inequalities to explore the attainability of optimal convergence rates for the local linear conditional mean estimator under algebraic NED conditions. Our results illustrate how the effective dimension affects assumptions of dependence. This finding refines the results of Jenish (2012) and extends the work of Hansen (2008), Vogt (2012), Chen and Christensen (2015) and Li, Lu, and Linton (2012).