Spatially adaptive density estimation by localised Haar projections

Gach, Florian; Nickl, Richard; Spokoiny, Vladimir

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 900-914.

A partir d’un échantillon d’une loi de densité $f_0:ℝ\to [0,\infty )$, nous construisons des estimateurs par ondelettes de Haar de $f_0$, dont les niveaux de résolution varient et sont construits à partir de tests localisés (comme dans l’article Lepski (Ann. Statist. 25 (1997) 927-947)). Nous montrons que ces estimateurs satisfont une inégalité oracle adaptive par rapport à la régularité potentiellement hétérogène de $f_0$, simultanément pour tout point $x$ dans un intervalle donné, en norme infinie. Les constantes de seuillage utilisées dans les procédures de test peuvent être choisies en pratique en supposant de manière idéalisée que la vraie densité est localement constante dans un voisinage du point $x$ considéré, pratique que nous justifions par un argument de théorie de l’information.

Given a random sample from some unknown density $f_0:ℝ\to [0,\infty )$ we devise Haar wavelet estimators for $f_0$ with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen and Spokoiny (Ann. Statist. 25 (1997) 927-947)). We show that these estimators satisfy an oracle inequality that adapts to heterogeneous smoothness of $f_0$, simultaneously for every point $x$ in a fixed interval, in sup-norm loss. The thresholding constants involved in the test procedures can be chosen in practice under the idealised assumption that the true density is locally constant in a neighborhood of the point $x$ of estimation, and an information theoretic justification of this practise is given.

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