Multivariate Kernel Estimators
The kernel estimate (4.4) can be generalized to the case of a multivariate regression function g: A → ℝ where A ⊂ ℝm, m ≥ 1. The proofs usually can be generalized from the univariate case without difficulty. There are, however, some genuinely new features in the multivariate situation. One is the sparsity of data, a problem that gets extremely worse with increasing dimension. Boundary effects are much more complicated. Choice of bandwidths and kernels requires further considerations, e.g. whether one should use the same bandwidth for all directions and whether one should use product kernels which are products of univariate functions.
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