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Let $f$ be a density on $[0,1]$ and let $X_1,X_2,\ldots$ be $\textit{iid}$ $f$-distributed. Also, let $f_n$ denote the kernel density estimator, i.e.

$$f_n(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\frac{x-X_i}{h_n}\right),$$

where $K$ is a kernel function (non-negative and integrates to 1) and $h_n$ is an appropriate bandwidth depending on $n$. I'm interested in finding the rate of convergence of the $L_1$ loss $$\mathbb{E}||f-f_n||_1 = \int_0^1|f(x)-f_n(x)| \text{d}x.$$ The existing literature is quite technical and I'm trying to understand what we need to impose on $f$ in order to get a result like $$\mathbb{E}||f-f_n||_1 = \mathcal{O}(n^{-s/(2s+1)})$$ when $f$ is $s$ times differentiable (among more restrictions). The literature I have been looking at does not seem to apply to my specific case or makes very technical assumptions (involving Besov norms). Searching for more available literature has not yet brought the desired result. So my question is, does there exists a book or article that clearly states a usable result for this setting? Or is it a matter of digging through a very dense topic?

These are the books I have been reading:

  • Devroye, L. and Györfi, L. Nonparametric density estimation: The $L_1$ View
  • Giné, E. and Nickl, R. Mathematical Foundations of Infinite-dimensional Statistical Models

NB. Any smoothness assumption on $f$ is allowed.

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  • $\begingroup$ The $\{L_p\}_{p \ge 1}$ spaces for a probability measure are nested: $L_{p} \subset L_q$ if $p \le q$. Here you are looking at the $L_1$ and $L_2$ spaces for the uniform prob. measure, so you have $\|g\|_1 \le \|g\|_2$. Any bound you have for the $L_2$ norm automatically gives a bound on the $L_1$. (If you are asking for the minimax risk in $L_1$ that requires to also provide a lower bound, which perhaps would be difficult and depend on the function class.) $\endgroup$
    – passerby51
    Commented Oct 4, 2017 at 15:37

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