Optimal rates for plug-in estimators of density level sets

By Philippe Rigollet, Régis Vert. 2006

135 citations recorded by the source corpus.

In the context of density level set estimation, we study the convergence of general plug-in methods under two main assumptions on the density for a given level λ. More precisely, it is assumed that the density (i) is smooth in a neighborhood of λ and (ii) has γ -exponent at level λ. Condition (i) ensures that the density can be estimated at a standard nonparametric rate and condition (ii) is similar to Tsybakov’s margin assumption which is stated for the classification framework. Under these assumptions, we derive optimal rates of convergence for plug-in estimators. Explicit convergence rates are given for plug-in estimators based on kernel density estimators when the underlying measure is the Lebesgue measure. Lower bounds proving optimality of the rates in a minimax sense when the density is Holder smooth are also provided.

Subfields: Artificial Intelligence, Control and Systems Engineering, Statistics and Probability

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Citations in the corpus

To Trust Or Not To Trust A Classifier (2018, background): “Our statistical guarantees are under the Hausdorff metric, which ensures a uniform guarantee over our estimator: it is a stronger notion of consistency than other common metrics [41, 43].”
Distribution Free Prediction Sets (2013, methodology): “The bottom right plot shows the estimator and prediction set based on the bandwidth whose corresponding conformal prediction set has the minimal Lebesgue measure.”
Estimation of extreme risk regions under multivariate regular variation (2011, methodology): “Two approaches are used, the plug-in approach using density estimation [see Báıllo, Cuesta-Albertos and Cuevas (2001) and Rigollet and Vert (2009)], and the excess mass approach [see Müller and Sawitzki (1991), Polonik (1995) and Tsybakov (1997)].”
A Review on Modal Clustering (2016, background): “Further references are Tsybakov (1997), Walther (1997), Cuevas et al. (2000), Baíllo et al. (2000), Baíllo et al. (2001), Baíllo (2003), Rigollet & Vert (2009) and Mason & Polonik (2009), and references therein, which delineate conditions, rates of convergence and further asymptotic results for a…”
Optimal weighted nearest neighbour classifiers (2011, result): “ensities. Marron (1983) proved that in a certain sense this is the minimax optimal rate. Further related work includes the literature on highest density region or level set estimation (Polonik, 1995; Rigollet and Vert, 2009; Samworth and Wand, 2010). Hall and Samworth (2005) and Biau and Devroye (20”
A Unifying Review of Deep and Shallow Anomaly Detection (2020, background): “We can formally express this objective as the problem of density level set estimation [214]–[217], which is equivalent to minimum volume set estimation [218]–[220] for the special case of density-based sets.”
Distribution‐free prediction bands for non‐parametric regression (2014, methodology): “Assumption 2 is related to the notion of the ‘γ-exponent’ condition that was introduced by Polonik (1995) and widely used in the density level set literature (Tsybakov, 1997; Rigollet and Vert, 2009).”
Distribution to Distribution Regression (2013, background): “The class I is the set of distribution Hk(1) with densities that are 1-smooth Hölder functions, as in (Rigollet & Vert, 2009).”