Note that the observations of should be centralized before
analyzed. Therefore, we assume
is centralized for ease of
exposition. In our proofs, we need the following conditions. (In
all our theorems, weaker conditions can be adopted at the expense
of much lengthier proofs.)
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(C1) is made only for the purpose of simplicity of proof. It can
be weakened to
for some
. Many time series models, including the autoregression
single-index model (Xia and Li; 1999), satisfy assumption (C1).
Assumption (C2) is also made for simplicity of proof. See, for
example, Härdle, Hall, and Ichimura (1993). The existence of finite moments is
sufficient. (C3) is needed for the uniform consistency of the
kernel smoothing methods. Assumption (C4) is needed for kernel
estimation of dependent data. Assumption (C5) is made to meet the
continuous requirement for kernel smoothing. The kernel
assumption (C6) is satisfied by most of the commonly used kernel
functions. For ease of exposition, we further assume
.