Asymptotic Statistics

Vaart, Aad van der

doi:10.1017/cbo9780511802256

Cited by 2,991 publications

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“…This implies that sup θ (q) Ψ M (θ(q)) − Ψ(θ(q)) (see equation (3)) converges to zero in probability since M/N A → I A < ∞. Using theorem 5.7 of Van der Vaart (1998) completes the proof.…”

Section: Discussionmentioning

confidence: 67%

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Counterfactual distributions with sample selection adjustments: Econometric theory and an application to the Netherlands

2009

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Several recent papers use the quantile regression decomposition method of Machado and Mata (2005) to analyze the gender gap in log wages across the distribution. Since employment rates often differ substantially by gender, sample selection is potentially a serious issue for such studies. To address this issue, we extend the Machado-Mata technique to account for selection. In addition, we prove that this procedure yields consistent and asymptotically normal estimates of the quantiles of the counterfactual distribution that it is designed to simulate.We illustrate our approach by analyzing the gender log wage gap between men and women who work full time in the Netherlands. Because the fraction of women working full time in the Netherlands is quite low, this is a case in which sample selection is clearly important. We find a positive selection of women into full-time work and find that about two thirds of this selection is due to observables such as education and experience with the remainder due to unobservables. Our decompositions show that the majority of the gender gap across the log wage distribution is due to differences between men and women in the distributions of returns to labor market characteristics rather than to differences in the distributions of the characteristics themselves.

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Section: Discussionmentioning

confidence: 67%

“…It is not difficult to show that this condition is satisfied. See Van der Vaart (1998) for more details.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Counterfactual distributions with sample selection adjustments: Econometric theory and an application to the Netherlands

2009

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“…Since our model is differentiable in quadratic mean, according to Theorem 7.2 of [28], under H 0 , the log likelihood ratio verifies…”

Section: Study Of the Score Process Under The Local Alternativementioning

confidence: 71%

An asymptotic test for Quantitative Trait Locus detection in presence of missing genotypes

Rabier¹

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…Therefore, it follows from Example 19.7 in [40] that the function class {K l (·, x) : x ∈ X } is VC(b l , a l , v l ) type class with b l ≤ C2 l , a l ≤ C2 l , and v l ≤ C, and so Lemma B.2 implies that K n,f is VC(b n , a n , v n ) type class with b n ≤ C2 lmax,n/2 , a n ≤ C2 2lmax,n , and v n ≤ C where l max,n = sup{L n }. Hence, the assumption that b 2 n σ 4 n K 4 n /n ≤ C 2 n −c 2 becomes 2 lmax,n (log 4 n)/n ≤ C 2 n −c 2 (with possibly different c 2 , C 2 ) as long as l max,n ≤ C log n for some C > 0.…”

Section: Supplemental Materialsmentioning

confidence: 86%

Anti-concentration and honest, adaptive confidence bands

Kato¹,

Chetverikov²,

Chernozhukov³

2013

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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.

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Asymptotic Statistics

Cited by 2,991 publications

References 0 publications

Counterfactual distributions with sample selection adjustments: Econometric theory and an application to the Netherlands

Counterfactual distributions with sample selection adjustments: Econometric theory and an application to the Netherlands

An asymptotic test for Quantitative Trait Locus detection in presence of missing genotypes

Anti-concentration and honest, adaptive confidence bands

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