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I'm new to kernel estimation methods and I've reading the paper "Fast Computation of Multivariate Kernel Estimators" by M. P. Wand.

Particulary on page 434, it says "Let $(X_1, Y_1), ..., (X_n, Y_n)$ be a sample of pairs, where $X_1, ..., X_n$ are $\mathbb{R^d}$-valued predictor variables having $d$-variate density $f$, and $Y_1, ..., Y_n$ are scalar response variables. The kernel density estimate of $f(x)$ and local polynomial kernel estimators of $m(x) = E(Y|X = x)$ depend on quantities of the form

$$\hat{s}_k(x) = \sum_{i=1}^n(X_i - x)^kK_h^P(X_i-x)$$ (and similar for $\hat{t_k}$)

where $k=(k_1, ..., k_d)$, and for a $d$-vector $u=(u_1, ..., u_d)$ the convention $u_k=(u_1^{k_1} ... u_d^{k_d})$ is used.

Then $k$ is used to calculate the kernel weights $\kappa$ which depend on $k, K$ and $h$ (the kernel and bandwidth), and $l$ (which is introduced later).

What do $k$ and $u$ mean here? It seems like it is something the reader is supposed to know because it comes out of nowhere.

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