Structural adaptation via Lp-norm oracle inequalities

By Alexander Goldenshluger, O. Lepski. 2007

66 citations recorded by the source corpus.

In this paper we study the problem of adaptive estimation of a multivari- ate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary L p-losses. These results are applied for adaptive estimation in the additive multi-index model.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/bd050c9cd2464414c0a79bc111f2758867a8c3e4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Yen-Chi Chen
2. Richard Nickl
3. Evarist Gin'e
4. O. Lepski
5. Anatoli B. Juditsky
6. Alexandre B. Tsybakov
7. Martin J. Wainwright
8. Karim Lounici

In Computational Theory and Mathematics

1. Yen-Chi Chen
2. Evarist Gin'e
3. Richard Nickl
4. O. Lepski
5. Anatoli B. Juditsky
6. Alexandre B. Tsybakov
7. Mengdi Wang
8. Guanghui Lan

In Statistics and Probability

1. Richard Nickl
2. Yen-Chi Chen
3. Evarist Gin'e
4. O. Lepski
5. Alexander Goldenshluger
6. I. Castillo
7. D. Chetverikov
8. Victor Chernozhukov

Citations in the corpus

• Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality (2010, background): “Define also For every h ∈ H let Then the selected bandwidth ˆ h and the corresponding kernel density estimator are defined by ˆ h := arg inf The selection rule (6)–(8) is a refinement of the one introduced recently in Goldenshluger and Lepski (2008, 2009) for the Gaussian white noise model.”
• NONPARAMETRIC ESTIMATION OF COMPOSITE FUNCTIONS (2009, background): “Examples of more complex commutative weight systems can be found in [8, 20].”
• A tutorial on kernel density estimation and recent advances (2017, methodology): “In addition to the above approaches, [35] proposes a method, known as the Lepski’s approach [36,37], that treats the bandwidth selection problem as a model selection problem and proposes a new criterion for selecting the smoothing bandwidth.”
• Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections (2008, result): “We should mention the results in Tsybakov [34], Golubev, Lepski and Levit [16] – who only considered Sobolev-type smoothness conditions – and [15].”
• CONFIDENCE BANDS IN DENSITY ESTIMATION (2010, background): “As mentioned above, Low (1997) shows some limitations for pointwise confidence intervals in density estimation. Our “generic” conditions circumvent his “pathologies.” The paper closest to the present one is Picard and Tribouley (2000) where pointwise adaptive confidence intervals are constructed in ”