Asymptotic equivalence of nonparametric regression and white noise

Brown, Lawrence D.; Low, Mark G.

doi:10.1214/aos/1032181159

Cited by 300 publications

(304 citation statements)

References 4 publications

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“…Of course, the previous remark is not a result of asymptotic equivalence of experiments, as defined by Le Cam (1972 and1986) and illustrated, for instance, by Brown andLow (1996) or Nussbaum (1996), among other examples. Our paralellism has some limitations: it holds up to constants, we restrict to loss functions of the form s −ŝ q 2 and h q (s,ŝ) (although this could easily be generalized) and to specific estimators, namely T -estimators.…”

Section: A Parallel With the White Noise Frameworkmentioning

confidence: 92%

Model selection via testing: an alternative to (penalized) maximum likelihood estimators

Birgé¹

2006

Annales de l'Institut Henri Poincare (B) Probability and Statis

189

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This paper is devoted to the description and study of a family of estimators, that we shall call T -estimators (T for tests), for minimax estimation and model selection. Their construction is based on former ideas about deriving estimators from some families of tests due to Le Cam (1973 and1975) and Birgé (1983, 1984a and and about complexity based model selection from Barron and Cover (1991).It is well-known that maximum likelihood estimators or, more generally, minimum contrast estimators do suffer from various weaknesses, and their penalized versions as well. In particular they are not robust and they require restrictive assumptions on both the models and the underlying parameter for the estimators to work correctly. Our method, which derives an estimator from many simultaneous tests between some probability balls in a suitable metric space, tends to solve many of these difficulties. Its robustness properties allow to deal with minimax estimation and model selection in a unified way, since bounding the minimax risk amounts to perform the method with a single, well-chosen, model. This results in simple bounds for the minimax risk solely based on some metric properties of the parameter space. Moreover the method applies to various statistical frameworks (we shall concentrate here on the i.i.d. and the Gaussian sequence settings only) and can handle essentially all types of models, linear or not, parametric and non-parametric, simultaneously. From these viewpoints, it is much more flexible than traditional methods. In particular, we shall be able to derive some adaptation results for density estimation over Besov balls that do not seem to be accessible to classical methods. The counterpart for these nice properties is the very high computational complexity of our construction which makes our estimators look more like theoretical than practical tools. 0 AMS 1991 subject classifications. Primary 62F35; secondary 62G05.

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Section: A Parallel With the White Noise Frameworkmentioning

confidence: 92%

Model selection via testing: an alternative to (penalized) maximum likelihood estimators

Birgé¹

2006

Annales de l'Institut Henri Poincare (B) Probability and Statis

189

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“…For the lower bounds we appeal to the equivalence between the regression and the Gaussian white noise model, as established by Brown and Low (1996), and consider merely the idealized observation model…”

Section: We Shall Use Throughout the Notationmentioning

confidence: 99%

Spectral calibration of exponential Lévy models

2006

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Abstract. We investigate the problem of calibrating an exponential Lévy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.

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“…The idea is to discretise the Gaussian shift experiment with a "step of discretisation" larger than 1/T . This method has already been used in Brown and Zhao [7] for proving the asymptotic equivalence between regression models with random and deterministic designs.…”

Section: Equivalence With Heteroskedastic Gaussian Regressionmentioning

confidence: 99%

Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case

Dalalyan

Reiß

2006

Probab. Theory Relat. Fields

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Abstract. Asymptotic local equivalence in the sense of Le Cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of Gaussian shift experiments. The nonparametric local neighbourhoods can be attained for any dimension, provided the regularity of the drift is sufficiently large. In addition, a heteroskedastic Gaussian regression experiment is given, which is also locally asymptotically equivalent and which does not depend on the centre of localisation. For one direction of the equivalence an explicit Markov kernel is constructed.

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Asymptotic equivalence of nonparametric regression and white noise

Cited by 300 publications

References 4 publications

Model selection via testing: an alternative to (penalized) maximum likelihood estimators

Model selection via testing: an alternative to (penalized) maximum likelihood estimators

Spectral calibration of exponential Lévy models

Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case

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