1  Multivariate nonparametric regression

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Suppose we observe data \((\mathbf{x}_1,Y_1),\dots,(\mathbf{x}_n,Y_n)\in [0,1]^p\times \mathbb{R}\), where \[ Y_i = m(\mathbf{x}_i) + \varepsilon_i, \] \(i = 1,\dots,n\), where \(m\) is an unknown function on \([0,1]^p\) and \(\varepsilon_1,\dots,\varepsilon_n\) are independent random variables with mean \(0\) and variance \(\sigma^2\).

As before, we will most of the time treat \(\mathbf{x}_1,\dots,\mathbf{x}_n\) as deterministic (fixed).

Our goal will be to estimate the unknown function \(m\).