EconStor Collection: cemmap working papers, Centre for Microdata Methods and Practice, Institute for Fiscal Studies (IFS)
http://hdl.handle.net/10419/64637
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Anticoncentration and honest, adaptive confidence bands
http://hdl.handle.net/10419/97423
Title: Anticoncentration and honest, adaptive confidence bands
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<br/>Authors: Chernozhukov, Victor; Chetverikov, Denis; Kato, Kengo
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<br/>Abstract: Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the classical SmirnovBickelRosenblatt (SBR) condition; see, for example, GinĂ© and Nickl (2010). 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 anticoncentration 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 nonparametetric 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 datadriven 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, nonconservative resolution levels via a Gaussian multiplier bootstrap method.

Convolution without independence
http://hdl.handle.net/10419/97422
Title: Convolution without independence
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<br/>Authors: Schennach, Susanne
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<br/>Abstract: Widely used convolutions and deconvolutions techniques traditionally rely on the assumption of independence, an assumption often criticized as being very strong. We observe that independence is, in fact, not necessary for the convolution theorem to hold. Instead, a much weaker notion, known as subindependence, is the appropriate necessary and sufficient condition. We motivate the usefulness of the subindependence concept by showing that is arguably as weak as a conditional mean assumption. We also provide an equivalent definition of subindependence that does not involve Fourier transforms and devise a constructive method to generate pairs of subindependent random variables.

Optimal uniform convergence rates for sieve nonparametric instrumental variables regression
http://hdl.handle.net/10419/97421
Title: Optimal uniform convergence rates for sieve nonparametric instrumental variables regression
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<br/>Authors: Chen, Xiaohong; Christensen, Timothy
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<br/>Abstract: We study the problem of nonparametric regression when the regressor is endogenous, which is an important nonparametric instrumental variables (NPIV) regression in econometrics and a difficult illposed inverse problem with unknown operator in statistics. We first establish a general upper bound on the supnorm (uniform) convergence rate of a sieve estimator, allowing for endogenous regressors and weakly dependent data. This result leads to the optimal supnorm convergence rates for spline and wavelet least squares regression estimators under weakly dependent data and heavytailed error terms. This upper bound also yields the supnorm convergence rates for sieve NPIV estimators under i.i.d. data: the rates coincide with the known optimal L2 norm rates for severely illposed problems, and are power of log (n) slower than the optimal L2 norm rates for mildly illposed problems. We then establish the minimax risk lower bound in supnorm loss, which coincides with our upper bounds on supnorm rates for the spline and wavelet sieve NPIV estimators. This supnorm rate optimality provides another justification for the wide application of sieve NPIV estimators. Useful results on weaklydependent random matricies are also provided.

Posterior inference in curved exponential families under increasing dimensions
http://hdl.handle.net/10419/97420
Title: Posterior inference in curved exponential families under increasing dimensions
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<br/>Authors: Belloni, Alexandre; Chernozhukov, Victor
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<br/>Abstract: This work studies the large sample properties of the posteriorbased inference in the curved exponential family under increasing dimension. The curved structure arises from the imposition of various restrictions on the model, such as moment restrictions, and plays a fundamental role in econometrics and others branches of data analysis. We establish conditions under which the posterior distribution is approximately normal, which in turn implies various good properties of estimation and inference procedures based on the posterior. In the process we also revisit and improve upon previous results for the exponential family under increasing dimension by making use of concentration of measure. We also discuss a variety of applications to highdimensional versions of the classical econometric models including the multinomial model with moment restrictions, seemingly unrelated regression equations, and single structural equation models. In our analysis, both the parameter dimension and the number of moments are increasing with the sample size.