This paper deals with the nonparametric estimation of the mean and variance functions of univariate time series data. We propose a nonparametric dimension reduction technique for both mean and variance functions of time series. This method does not require any model specification and instead we seek directions in both the mean and variance functions such that the conditional distribution of the current observation given the vector of past observations is the same as that of the current observation given a few linear combinations of the past observations without loss of inferential information. The directions of the mean and variance functions are estimated by maximizing the Kullback–Leibler distance function. The consistency of the proposed estimators is established. A computational procedure is introduced to detect lags of the conditional mean and variance functions in practice. Numerical examples and simulation studies are performed to illustrate and evaluate the performance of the proposed estimators.

中文翻译：

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时间序列中条件均值和方差函数的降维

本文处理单变量时间序列数据的均值和方差函数的非参数估计。我们为时间序列的均值和方差函数提出了一种非参数降维技术。这种方法不需要任何模型规范，而是我们在均值和方差函数中寻找方向，使得给定过去观察向量的当前观察的条件分布与给定几个线性组合的当前观察的条件分布相同过去的观察而不丢失推理信息。通过最大化 Kullback-Leibler 距离函数来估计均值和方差函数的方向。建立了建议的估计量的一致性。在实践中引入了一种计算程序来检测条件均值和方差函数的滞后。进行了数值示例和模拟研究，以说明和评估所提出的估计器的性能。