Global uniform risk bounds for wavelet deconvolution estimators

Lounici, Karim; Nickl, Richard

doi:10.1214/10-aos836

Cited by 78 publications

(75 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our proof of sup-norm lower bound for NPIV models is similar to that of Chen and Reiss (2011) for L 2 -norm lower bound. Similar sup-norm lower bounds for density deconvolution were recently obtained by Lounici and Nickl (2011).…”

Section: Lower Boundssupporting

confidence: 80%

Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression

Chen¹,

Christensen²

2017

View full text Add to dashboard Cite

This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function h 0 and its functionals. First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series 2SLS) estimators of h 0 and its derivatives. Second, we derive a lower bound that describes the best possible (minimax) sup-norm rates of estimating h 0 and its derivatives, and show that the sieve NPIV estimator can attain the minimax rates when h 0 is approximated via a spline or wavelet sieve. Our optimal sup-norm rates surprisingly coincide with the optimal root-mean-squared rates for severely ill-posed problems, and are only a logarithmic factor slower than the optimal root-mean-squared rates for mildly ill-posed problems. Third, we use our sup-norm rates to establish the uniform Gaussian process strong approximations and the score bootstrap uniform confidence bands (UCBs) for collections of nonlinear functionals of h 0 under primitive conditions, allowing for mildly and severely ill-posed problems. Fourth, as applications, we obtain the first asymptotic pointwise and uniform inference results for plug-in sieve t-statistics of exact consumer surplus (CS) and deadweight loss (DL) welfare functionals under low-level conditions when demand is estimated via sieve NPIV. Empiricists could read our real data application of UCBs for exact CS and DL functionals of gasoline demand that reveals interesting patterns and is applicable to other markets.

show abstract

Section: Lower Boundssupporting

confidence: 80%

Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression

Chen¹,

Christensen²

2017

View full text Add to dashboard Cite

show abstract

“…If α = 0 the same bound was obtained in Goldenshluger and Lepski (2014) and the asymptotics found in the first assertion is the minimax rate of convergence, see Lepski (2013). In the univariate case d = 1, if α = 1, and r = ∞ (Hölder class) the first assertion of the theorem was proved in Lounici and Nickl (2011). Nikol'skii spaces, Nikol'skii (1977), Section 6.9, the condition τ (∞) > 0 guarantees that all the functions belonging to …”

Section: Bounds For Thesupporting

confidence: 58%

“…They also establish an upper bound under the additional assumptions that 1 ≤ r ≤ 2, µ < r 2−r (β + 1 2 − 1 r ) and f has a fixed compact support. Lounici and Nickl (2011) examined the case of the L ∞ -loss. They obtain a lower and an upper bound for the minimax risk over the class of Holder spaces Σ = B ( β, ∞, ∞, L ) with β > 0, but they remark that the results can be generalized to the class of Besov spaces Σ = B ( β, r, q, L ) .…”

Section: Introductionmentioning

confidence: 99%

Lower bounds in the convolution structure density model

Lepski¹,

Willer²

2017

Bernoulli

View full text Add to dashboard Cite

Abstract:The aim of the paper is to establish asymptotic lower bounds for the minimax risk in two generalized forms of the density deconvolution problem. The observation consists of an independent and identically distributed (i.i.d.) sample of n random vectors in R d . Their common probability distribution function p can be written aswhere f is the unknown function to be estimated, g is a known function, α is a known proportion, and ⋆ denotes the convolution product. The bounds on the risk are established in a very general minimax setting and for moderately ill posed convolutions. Our results show notably that neither the ill-posedness nor the proportion α play any role in the bounds whenever α ∈ [0, 1), and that a particular inconsistency zone appears for some values of the parameters. Moreover, we introduce an additional boundedness condition on f and we show that the inconsistency zone then disappears.AMS 2000 subject classifications: 62G05, 62G20.

show abstract

“…However, to establish the validity of bootstrap confidence bands, researchers relied on the existence of continuous limit distributions of normalized suprema of original studentized processes. In the deconvolution density estimation problem, [28] considered confidence bands without using Gaussian approximation. In the current density estimation problem, their idea reads as bounding the deviation probability of f n −E[f n (·)] ∞ by using Talagrand's [38] inequality and replacing the expected supremum by the Rademacher average.…”

Section: Introductionmentioning

confidence: 99%

Anti-concentration and honest, adaptive confidence bands

Kato¹,

Chetverikov²,

Chernozhukov³

2013

View full text Add to dashboard Cite

Abstract. Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the the classical Smirnov-Bickel-Rosenblatt (SBR) condition; see, for example, . This condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process (equivalently, for the supremum of a Gaussian process with the same covariance function as that of the studentized empirical process). The principal contribution of this paper is to remove the need for this classical condition. We show that a considerably weaker sufficient condition is derived from an anti-concentration property of the supremum of the approximating Gaussian process, and we derive an inequality leading to such a property for separable Gaussian processes. We refer to the new condition as a generalized SBR condition. Our new result shows that the supremum does not concentrate too fast around any value.We then apply this result to derive a Gaussian multiplier bootstrap procedure for constructing honest confidence bands for nonparametric density estimators (this result can be applied in other nonparametric problems as well). An essential advantage of our approach is that it applies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist (or is unknown). This is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data-driven fashion, which is needed for adaptive constructions of the confidence bands. Furthermore, our approach is asymptotically honest at a polynomial rate -namely, the error in coverage level converges to zero at a fast, polynomial speed (with respect to the sample size). In sharp contrast, the approach based on extreme value theory is asymptotically honest only at a logarithmic rate -the error converges to zero at a slow, logarithmic speed. Finally, of independent interest is our introduction of a new, practical version of Lepski's method, which computes the optimal, non-conservative resolution levels via a Gaussian multiplier bootstrap method.

show abstract

Global uniform risk bounds for wavelet deconvolution estimators

Cited by 78 publications

References 43 publications

Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression

Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression

Lower bounds in the convolution structure density model

Anti-concentration and honest, adaptive confidence bands

Contact Info

Product

Resources

About