### Citations

195 |
Measure Theory and Fine
- Evans, Gariepy
- 1992
(Show Context)
Citation Context ...he book by Bosq and Lecoutre [17], chapter 4 and 5. From now on, we denote by ∂A the boundary of any subset A ⊂ Rd. Besides, we introduce H the (d− 1)-dimensional Hausdorff measure (Evans and Gariepy =-=[23]-=-). Recall that H agrees with ordinary (k - 1)-dimensional surface area on nice sets (Proposition A.1 in [1]). Finally, we set K̃ = ∫ K2dλ. 9 ha l-0 06 74 19 7,sv er sio ns3s- 8sO cts2 01 2 4.1.1. Pr... |

120 |
Measuring mass concentrations and estimating density contour clusters: an excess mass approach
- Polonik
- 1995
(Show Context)
Citation Context ...t) = {x ∈ Rd : r̂n(x) > t}. Most of the research works on the estimation of level sets concern the density function. One can cite the works of Cadre [1], Cuevas and Fraiman [2], Hartigan [3], Polonik =-=[4]-=-, Tsybakov [5], Walther [6]. This large number of works on this subject is motivated by the high number of possible applications. Estimating these level sets can be useful in mode estimation (Müller a... |

96 |
The Accuracy of the Gaussian Approximation to the Sum of Independent Variates
- Berry
- 1941
(Show Context)
Citation Context ... ( 1 n n∑ i=1 ( Zi(x, t)− EZ(x, t) ) < −EZ(x, t) ) = P (√ n Vn(x, t) 1 n n∑ i=1 ( Zi(x, t)− EZ(x, t) ) < tn(x) ) . 12 ha l-0 06 74 19 7,sv er sio ns3s- 8sO cts2 01 2 Then, the Berry-Esseen inequality =-=[24]-=- gives us |P(rn(x) < t)−Φ(tn(x))| ≤ c√ nVn(x, t)3 E |(Y − t)Kh(x−X)− E(Y − t)Kh(x−X)|3 . (6) Finally, under Assumptions A1 and A3, we have (see for example Bosq and Lecoutre [17]) sup x∈L(t−) |(Y − t)... |

78 |
On nonparametric estimation of density level sets
- Tsybakov
- 1997
(Show Context)
Citation Context ... r̂n(x) > t}. Most of the research works on the estimation of level sets concern the density function. One can cite the works of Cadre [1], Cuevas and Fraiman [2], Hartigan [3], Polonik [4], Tsybakov =-=[5]-=-, Walther [6]. This large number of works on this subject is motivated by the high number of possible applications. Estimating these level sets can be useful in mode estimation (Müller and Stawitzki [... |

65 |
Excess mass estimates and tests of multimodality
- Muller, Sawitzki
- 1991
(Show Context)
Citation Context ...], Walther [6]. This large number of works on this subject is motivated by the high number of possible applications. Estimating these level sets can be useful in mode estimation (Müller and Stawitzki =-=[7]-=-, Polonik [4]), or in clustering (Biau, Cadre and Pelletier [8], Cuevas, Febrero and Fraiman [9, 10]). In particular, Biau, Cadre and Pelletier [8] use an estimator of the level sets of the density fu... |

50 |
Estimation of a convex density contour in two dimensions
- Hartigan
- 1987
(Show Context)
Citation Context ...e L(t) by Ln(t) = {x ∈ Rd : r̂n(x) > t}. Most of the research works on the estimation of level sets concern the density function. One can cite the works of Cadre [1], Cuevas and Fraiman [2], Hartigan =-=[3]-=-, Polonik [4], Tsybakov [5], Walther [6]. This large number of works on this subject is motivated by the high number of possible applications. Estimating these level sets can be useful in mode estimat... |

45 | Kernel estimation of regression functions. Smoothing techniques for curve estimation - Gasser, Müller - 1979 |

24 |
Nonparametric estimation of regression level sets
- Cavalier
- 1997
(Show Context)
Citation Context ... minimax rates (for different smoothness classes) for estimators based on recursive dyadic partitions. Scott and Davenport [14] use a cost sensitive approach and a different measure of risk. Cavalier =-=[15]-=- and Polonik and Wang [16] used estimators based on the maximization of the excess mass which was introduced by Müller and Sawitzki [7] and Hartigan [3]. Cavalier demonstrated asymptotic minimax rate ... |

24 |
Plug-in estimation of general level sets
- Cuevas, González-Manteiga, et al.
(Show Context)
Citation Context ... λ({r = t}) = 0 (where λ stands for the Lebesgue measure), a first consistency result can be trivialy obtained from a slight modification of Theorem 3 by Cuevas, González-Manteiga and Rodríguez-Casal =-=[19]-=- and the consistency properties of the kernel estimator. Proposition 2.1. Under Assumption A0, if K is bounded, integrable, with compact support and Lipschitz, and if h→ 0 and nhd/ log n→∞, then Eλ ( ... |

21 | Kernel estimation of density level sets
- Cadre
- 2006
(Show Context)
Citation Context ...stimator r̂n of r, in order to estimate L(t) by Ln(t) = {x ∈ Rd : r̂n(x) > t}. Most of the research works on the estimation of level sets concern the density function. One can cite the works of Cadre =-=[1]-=-, Cuevas and Fraiman [2], Hartigan [3], Polonik [4], Tsybakov [5], Walther [6]. This large number of works on this subject is motivated by the high number of possible applications. Estimating these le... |

21 |
Cluster analysis: a further approach based on density estimation
- Cuevas, Febrero, et al.
- 2001
(Show Context)
Citation Context ...ssible applications. Estimating these level sets can be useful in mode estimation (Müller and Stawitzki [7], Polonik [4]), or in clustering (Biau, Cadre and Pelletier [8], Cuevas, Febrero and Fraiman =-=[9, 10]-=-). In particular, Biau, Cadre and Pelletier [8] use an estimator of the level sets of the density function to determine the number of clusters. The same applications are possible with the regression f... |

21 | Minimax optimal level set estimation
- Willett, Nowak
- 2007
(Show Context)
Citation Context ...is. Despite the many potential applications, the estimation of the level sets of the regression function has not been widely studied. Müller [12] mentioned it briefly in his survey. Willett and Nowak =-=[13]-=- obtained minimax rates (for different smoothness classes) for estimators based on recursive dyadic partitions. Scott and Davenport [14] use a cost sensitive approach and a different measure of risk. ... |

16 | Stability of density-based clustering
- Rinaldo, Singh, et al.
(Show Context)
Citation Context ...set estimation is still an interesting and opened question. For this, we could first think of the adaptation of method used for density level sets estimation like Rinaldo, Singh, Nugent and Wasserman =-=[21]-=- or Samworth and Wand [22] for example. 4. Proofs This section is dedicated to the proof of Theorem 2.1. From now on, c is a non-negative constant, which value may change from line to line. 4.1. Proof... |

14 |
A graph-based estimator of the number of clusters
- Biau, Cadre, et al.
- 2007
(Show Context)
Citation Context ...otivated by the high number of possible applications. Estimating these level sets can be useful in mode estimation (Müller and Stawitzki [7], Polonik [4]), or in clustering (Biau, Cadre and Pelletier =-=[8]-=-, Cuevas, Febrero and Fraiman [9, 10]). In particular, Biau, Cadre and Pelletier [8] use an estimator of the level sets of the density function to determine the number of clusters. The same applicatio... |

14 | Regression level set estimation via cost-sensitive classification
- Scott, Davenport
- 2007
(Show Context)
Citation Context ...ller [12] mentioned it briefly in his survey. Willett and Nowak [13] obtained minimax rates (for different smoothness classes) for estimators based on recursive dyadic partitions. Scott and Davenport =-=[14]-=- use a cost sensitive approach and a different measure of risk. Cavalier [15] and Polonik and Wang [16] used estimators based on the maximization of the excess mass which was introduced by Müller and ... |

11 | Expression profiles of osteosarcoma that can predict response to chemotherapy
- Man, Chintagumpala, et al.
- 2005
(Show Context)
Citation Context ...n also find a lot of applications. For instance, the severity of the cancer is characterized by a variable Y which directly impacts the choice of standard or aggressive chemotherapy. For osteosarcoma =-=[11]-=-, Y is the percent necrosis in the tumor after a first round of treatment. If Y > 0.9 (this threshold has been fixed by experts and is now the convention), the aggressive chemotherapy will be chosen. ... |

11 |
Convergent estimators for the L1-median of a Banach-valued random variable
- Cadre
- 2001
(Show Context)
Citation Context ...) 1 d and (n log n) −1 d+4 . Moreover, if we take h = O((n log n) −1 d+4 ), we get √ nhd = O ( n 2 d+4 (log n) d 2(d+4) ) = O ( n1/3 (log n)1/6 ) with d = 2, that is a rate of the same order as Cadre =-=[20]-=- in the density case. • A remaining and crucial problem is the research of an optimal bandwidth h for our estimator. Indeed, if they are already results in the literature about an optimal bandwidth fo... |

10 |
Estimating the number of clusters. The Canadian
- Cuevas, Febrero, et al.
- 2000
(Show Context)
Citation Context ...ssible applications. Estimating these level sets can be useful in mode estimation (Müller and Stawitzki [7], Polonik [4]), or in clustering (Biau, Cadre and Pelletier [8], Cuevas, Febrero and Fraiman =-=[9, 10]-=-). In particular, Biau, Cadre and Pelletier [8] use an estimator of the level sets of the density function to determine the number of clusters. The same applications are possible with the regression f... |

10 | Asymptotics and optimal bandwidth selection for highest density region estimation
- Samworth, Wand
- 2010
(Show Context)
Citation Context ... interesting and opened question. For this, we could first think of the adaptation of method used for density level sets estimation like Rinaldo, Singh, Nugent and Wasserman [21] or Samworth and Wand =-=[22]-=- for example. 4. Proofs This section is dedicated to the proof of Theorem 2.1. From now on, c is a non-negative constant, which value may change from line to line. 4.1. Proof of Theorem 2.1 In this pr... |

6 |
The Excess Mass Approach in Statistics, Beiträge zur Statistik
- Müller
- 1993
(Show Context)
Citation Context ... in most practical situations, particularly in image analysis. Despite the many potential applications, the estimation of the level sets of the regression function has not been widely studied. Müller =-=[12]-=- mentioned it briefly in his survey. Willett and Nowak [13] obtained minimax rates (for different smoothness classes) for estimators based on recursive dyadic partitions. Scott and Davenport [14] use ... |

4 |
Théorie de l'Estimation Fonctionnelle, Ecole Nationale de la Statistique et de l'Administration Economique et Centre d'Etudes des Programmes Economiques, Economica
- Bosq, Lecoutre
- 1987
(Show Context)
Citation Context ...ed by rn(x) = { ϕn(x)/fn(x) if fn(x) 6= 0 0 otherwise. The properties of this estimator are already well studied in the litterature. For instance, the interesting reader can look at Bosq and Lecoutre =-=[17]-=- or Gasser and Müller [18]. Under the assumption A0 There exists t− < t such that L(t−) is compact. Besides, λ({r = t}) = 0 (where λ stands for the Lebesgue measure), a first consistency result can be... |

3 | A plug-in approach to support estimation. The Annals of Statistics, 25(6):2300–2312 - Cuevas, Fraiman - 1997 |

2 |
Granulometric smoothing, The Annals of Statistics 25 (6
- Walther
- 1997
(Show Context)
Citation Context ... Most of the research works on the estimation of level sets concern the density function. One can cite the works of Cadre [1], Cuevas and Fraiman [2], Hartigan [3], Polonik [4], Tsybakov [5], Walther =-=[6]-=-. This large number of works on this subject is motivated by the high number of possible applications. Estimating these level sets can be useful in mode estimation (Müller and Stawitzki [7], Polonik [... |

2 |
Estimation of regression contour clusters: an application of the excess mass approach to regression, Journal of multivariate analysis 94
- Polonik, Wang
- 2005
(Show Context)
Citation Context ...ent smoothness classes) for estimators based on recursive dyadic partitions. Scott and Davenport [14] use a cost sensitive approach and a different measure of risk. Cavalier [15] and Polonik and Wang =-=[16]-=- used estimators based on the maximization of the excess mass which was introduced by Müller and Sawitzki [7] and Hartigan [3]. Cavalier demonstrated asymptotic minimax rate of convergence for piecewi... |