Adaptive estimation of linear functionals in the convolution model and applications
By Cristina Butucea, Fabienne Comte. 2009
60 citations recorded by the source corpus.
We consider the model Zi = Xi + "i for i.i.d. Xi’s and "i’s and independent sequences (Xi)i2N and ("i)i2N. The density of " is assumed to be known whereas the one of X1 denoted by g is unknown. Our aim is to study the estimation of linear functionals of g, h ,gi for a known function . We propose a general estimator of h ,gi and study the rate of convergence of its quadratic risk in function of the smoothness of g, f" and . Dierent dependency contexts are also considered. An adaptive estimator is then proposed, following a method studied by Laurent et al. [23] in another context. The quadratic risk of this estimator is studied. The results are applied to adaptive pointwise deconvolution, in which context losses in the adaptive rates are shown to be optimal in the minimax sense. They are also applied to pointwise Laplace transform estimation in the standard context and in the context of the stochastic volatility model. Estimation in the context of ARCH-type models lastly illustrates the method. MSC 2000 Subject Classifications. 62G07 - 62M05.
Subfields: Finance, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/c1ae3b8edadef1a395615a5d759cf36820507116
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In Statistics and Probability
- V. Zanten
- Van der Vaart
- I. Castillo
- Alexander Goldenshluger
- Anatoli B. Juditsky
- Itai Dattner
- B. Knapik
- Kolyan Ray
Citations in the corpus
- ON DECONVOLUTION OF DISTRIBUTION FUNCTIONS (2010, background): “Note also that [3] considered a general problem of optimal and adaptive estimation of linear functionals (f ) = ∫∞ −∞ φ(t)f (t)dt in the model (1).”
- Bayesian Recovery of the Initial Condition for the Heat Equation (2011, background): “The minimax rate over a ball in the Sobolev space Sβ is known to be bounded above by ( log n )−(β+q)/2 (for the case of q = −1/2 see Goldenshluger, 1999, and for general q in a closely related model see Butucea and Comte, 2009). In view of Theorem 2.1, it is not surprising that exponential priors yi”