An Improved Bernstein-type Inequality for C-Mixing-type Processes and Its Application to Kernel Smoothing
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There are many processes, particularly dynamic systems, that cannot be described as strong mixing processes. <cit.> introduced a new mixing coefficient called C-mixing, which includes a large class of dynamic systems. Based on this, <cit.> obtained a Bernstein-type inequality for a geometric C-mixing process, which, modulo a logarithmic factor and some constants, coincides with the standard result for the iid case. In order to honor this pioneering work, we conduct follow-up research in this paper and obtain an improved result under more general preconditions. We allow for a weaker requirement for the semi-norm condition, fully non-stationarity, non-isotropic sampling behavior. Our result covers the case in which the index set of processes lies in 𝐙^d+ for any given positive integer d. Here 𝐙^d+ denotes the collection of all nonnegative integer-valued d-dimensional vector. This setting of index set takes both time and spatial data into consideration. For our application, we investigate the theoretical guarantee of multiple kernel-based nonparametric curve estimators for C-Mixing-type processes. More specifically we firstly obtain the L^∞-convergence rate of the kernel density estimator and then discuss the attainability of optimality, which can also be regarded as an upate of the result of <cit.>. Furthermore, we investigate the uniform convergence of the kernel-based estimators of the conditional mean and variance function in a heteroscedastic nonparametric regression model. Under a mild smoothing condition, these estimators are optimal. At last, we obtain the uniform convergence rate of conditional mode function.
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