Second Order Properties of Nearest Neighbor Regression with Uniform Weighting Function for Strongly Mixing Processes

We examine the statistical properties of nearest neighbor regression function estimation beyond the classical i.i.d assumption. Second-order properties of this estimator for uniformly mixing processes have been derived in previous studies. Nevertheless, uniform mixing is a very strong form of dependence that is difficult to achieve. In contrast, strong mixing conditions are satisfied by a broad class of stochastic processes commonly encountered in theoretical and applied time series modeling. This paper focuses on the analysis of the second-order properties of the nearest neighbor regression estimator under strong mixing dependence measure. Under appropriate regularity conditions, including strong mixing assumptions and smoothness of the underlying density functions, we derive expressions for both the bias and the quadratic mean squared error (QMSE) of the nearest neighbor regression estimator with a uniform weighting scheme for estimating the unknown conditional mean function. Our results demonstrate that the QMSE of the nearest neighbor estimator attains the minimax-optimal rate for estimating a p-smooth regression function in a d-dimensional embedding space. The theoretical analysis integrates the dependence structure specific to strongly mixing processes with the geometric characteristics of k-nearest neighborhoods. This combination enables the identification of an optimal choice for the number of nearest neighbors that effectively balances the trade-off between bias and variance. Overall, these findings provide a rigorous theoretical foundation for the application of nearest neighbor regression methods to short-range dependent data. Furthermore, the explicit bias and variance characterizations lay the groundwork for establishing asymptotic normality, thereby enabling the construction of valid confidence intervals and supporting reliable statistical inference in dependent data settings.