GLOBAL UNIFORM RISK BOUNDS FOR WAVELET DECONVOLUTION ESTIMATORS

By Karim Lounici, Richard Nickl. 2011

89 citations recorded by the source corpus.

We consider the statistical deconvolution problem where one observes n replications from the model Y = X + ∈, where X is the unobserved random signal of interest and ∈ is an independent random error with distribution ϕ. Under weak assumptions on the decay of the Fourier transform of ϕ, we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators f n for the density f of X, where f :R → R is assumed to be bounded. We then derive lower bounds for the minimax sup-norm risk over Besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds. We further show that linear estimators adapt to the unknown smoothness of f if the Fourier transform of ϕ decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if ϕ decays polynomially. We also analyze the case where f is a "supersmooth"/analytic density. We finally show how our results and recent techniques from Rademacher processes can be applied to construct global confidence bands for the density f.

Subfields: Artificial Intelligence, Oncology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d3b4a405206594e09a04bd9fa8b80b0b2dde6073

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In Artificial Intelligence

In Oncology

In Statistics and Probability

Citations in the corpus

Gaussian approximation of suprema of empirical processes (2012, background): “Finally, we mention that there are alternative, yet more conservative, approaches on construction of confidence bands based on nonasymptotic concentration inequalities (and not on Gaussian approximation); see [39] and [32].”
Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression (2015, unknown): “Similar sup-norm lower bounds for density deconvolution were recently obtained by Lounici and Nickl (2011).”
Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞ (2011, methodology): “For the bias, we take Pn to be the support of Π and consider a Meyer-wavelet basis and the wavelet projection onto it, with 2Jn = c(logn)2κ, where c is a large enough constant that depends on inf{σ :σ ∈ supp(G)}, and apply Proposition 4 in [25] with s= 2 and suitable c̃0, to see that ‖KJn(pF,σ)−pF,σ”
Anti-concentration and honest, adaptive confidence bands (2013, methodology): “The authors would like to thank Emre Barut, Enno Mammen and Richard Nickl for very helpful discussions.”
On adaptive inference and confidence bands (2011, methodology): “We state a version that follows from Theorem 4 in [25] for T = R.”
Bayesian inverse problems with non-conjugate priors (2012, methodology): “Non-conjugate Bayesian inverse problems 2519 In this setting, unless the prior satisfies an analytic smoothness condition, the bias of the linear estimator dominates its variance [5, 24] and consequently the minimum of the prior smoothness and the unknown true smoothness determines the rate.”