Suppose that $X_1, X_2, \dots, X_n$ forms an independent and identically distributed sample from some $d$-dimensional probability distribution with unknown probability density function $f$. Let $x$ be some point in the interior of $f$'s support. The goal is to estimate $f$ based on the observed sample using some nonparametric estimator $\hat f$. Nonparametric estimators suffer from the curse of dimensionality; Stone [1] is often given as a reference to the claim that the optimal rate of convergence for any nonparametric estimator is $\mathcal O\big(n^{-\frac{p}{2p+d}}\big)$, where $p$ is the degree of smoothness. I don't understand how Stone's setting applies to a general class of nonparametric estimators. The setting stated in his paper appears very artificial. I would be glad if someone could shed some light on this.
Disclaimer: I have asked the same question an MathSE, but since it was not answered there, I hope to find an answer here.
