On the best rate of adaptive estimation in some inverse problems

By Alexandre B. Tsybakov. 2000

79 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9c62d1ad40d4772b4f7ec6160ed78b802ce9e57d

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In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Rates of convergence for nonparametric deconvolution (2006, background): “hat these original rates have the property to decrease faster than any logarithmic function. Moreover, they are optimal in the cases where the corresponding lower bounds are known, i.e r = s = 1 (see [11]) and r < s (see[2]). We can also remark that, given the complexity of these rates, it is wot”
Sharp Optimality in Density Deconvolution with Dominating Bias. II (2008, background): “To our knowledge, except for the case r = s = 1 treated by Golubev and Khasminskii (2001), Tsybakov (2000) and Cavalier, Golubev, Lepski and Tsybakov (2003), such lower bounds are not available even for the Gaussian white noise (or sequence space) deconvolution model, although some upper bounds are…”
Geometry of linear ill-posed problems in variable Hilbert scales Inverse Problems 19 789-803 (2003, background): “Since then many authors have adopted this approach towards various applications, we mention [27] and [4].”
Penalized contrast estimator for adaptive density deconvolution (2006, background): “This is shown in the special case 0 < r < δ and s = 0, in Butucea and Tsybakov (2004), where sharp minimax results are stated. This is also shown by our results when 0 < δ ≤ r and when 0 < r < δ, s 6= 0 (see Sections 3.4 and 4.2). For instance, when 0 < δ/r ≤ 1/2, according to (10) and Sections 3.3 ”
Comparing parameter choice methods for regularization of ill-posed problems (2011, background): “Since then it has been developed further for regularization of linear inverse problems [50, 135, 105, 106, 18] and nonlinear inverse problems [12, 13] in deterministic and stochastic settings.”
Sharp optimality for density deconvolution with dominating bias (2004, background): “To our knowledge, except for the case r = s = 1 treated by Golubev and Khasminskii (2001), Tsybakov (2000) and Cavalier, Golubev, Lepski and Tsybakov (2003), such lower bounds are not available even for the Gaussian white noise (or sequence space) deconvolution model, although some upper bounds are…”
Deconvolution of supersmooth densities with smooth noise (2004, methodology): “Deconvolution densities in the class of analytic functions (T = 1) mixed with Cauchy type noise were estimated by Tsybakov (2000), in a discrete model.”