Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cube root weak convergence of empirical estimators of a density level
  set   Given $n$ independent random vectors with common density $f$ on
$\mathbb{R}^d$, we study the weak convergence of three empirical-measure based
estimators of the convex $\lambda$-level set $L_\lambda$ of $f$, namely the
excess mass set, the minimum volume set and the maximum probability set, all
selected from a class of convex sets $\mathcal{A}$ that contains $L_\lambda$.
Since these set-valued estimators approach $L_\lambda$, even the formulation of
their weak convergence is non-standard. We identify the joint limiting
distribution of the symmetric difference of $L_\lambda$ and each of the three
estimators, at rate $n^{-1/3}$. It turns out that the minimum volume set and
the maximum probability set estimators are asymptotically indistinguishable,
whereas the excess mass set estimator exhibits "richer" limit behavior.
Arguments rely on the boundary local empirical process, its cylinder
representation, dimension-free concentration around the boundary of
$L_\lambda$, and the set-valued argmax of a drifted Wiener process.

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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: cs