Posterior inference in curved exponential families under increasing dimensions

Belloni, Alexandre; Chernozhukov, Victor

doi:10.1920/wp.cem.2013.6813

Cited by 2 publications

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“…The latter papers discuss applications to large contingency tables where the minimal cell expectations do not converge to infinity. Exponential families with increasing dimension were studied in Portnoy (1988) and Belloni and Chernozhukov (2012). For linear and log-linear models, Mammen (1989) and Sauermann (1989) showed consistency of bootstrap for linear contrasts under conditions where the normal approximation fails because of bias effects.…”

Section: Dimension Asymptoticsmentioning

confidence: 99%

Generalized Jackknife Estimators of Weighted Average Derivatives

Cattaneo

Crump

Jansson

2013

Journal of the American Statistical Association

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With the aim of improving the quality of asymptotic distributional approximations for nonlinear functionals of nonparametric estimators, this article revisits the large-sample properties of an important member of that class, namely a kernel-based weighted average derivative estimator. Asymptotic linearity of the estimator is established under weak conditions. Indeed, we show that the bandwidth conditions employed are necessary in some cases. A bias-corrected version of the estimator is proposed and shown to be asymptotically linear under yet weaker bandwidth conditions. Implementational details of the estimators are discussed, including bandwidth selection procedures. Consistency of an analog estimator of the asymptotic variance is also established. Numerical results from a simulation study and an empirical illustration are reported. To establish the results, a novel result on uniform convergence rates for kernel estimators is obtained. The online supplemental material to this article includes details on the theoretical proofs and other analytic derivations, and further results from the simulation study.

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Section: Dimension Asymptoticsmentioning

confidence: 99%

Generalized Jackknife Estimators of Weighted Average Derivatives

Cattaneo

Crump

Jansson

2013

Journal of the American Statistical Association

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confidence: 99%

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Mammen

2013

Journal of the American Statistical Association

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are to be congratulated for an interesting article with a new point of view on semiparametrics. Their nonstandard way to look at semiparametric estimation problems is very innovative and it is motivating for further research.The article studies what happens if one goes beyond the border of standard asymptotics. For a specific example, the article discusses a semiparametric estimation problem, where the nonparametric estimator has a poorer asymptotic performance than required from classical semiparametric theory. This is an important problem, in the concrete setting of the article and also in general theory. Often, in semiparametrics, assumptions are made on the nonparametric estimator that are not realistic. An example would be higher dimensional nonparametric regression functions where higher order smoothness assumptions are made that allow o P (n −1/4 ) convergence of the nonparametric estimator. There are some concerns in nonparametrics about the sense of such higher order smoothness conditions for moderate sample sizes, see, for example, Marron and Wand (1992). It is natural to argue that also in semiparametric contexts it is questionable if these higher order assumptions make sense. This motivates an asymptotic framework in semiparametrics, where such assumptions are avoided and where this problem is not neglected in the asymptotic limit. That is exactly what the authors of this article have done. I think that the article addresses a central question of mathematical statistics.As mentioned in the article, the discussions of the article are related to recent work of L. Li, J. Robins, E. Tchetgen, and A. van der Vaart, but a different point of view is taken here. It is assumed that the bias of the nonparametric estimator is negligible and does not influence the first-order asymptotics of the parametric estimator. Then the asymptotics of the parametric part is only affected by the stochastic part of the nonparametric estimator. As was shortly mentioned in the article, this relates the article to discussions on high-dimensional parametric models. Nonparametric regression can be interpreted as parametrics with increasing dimension. Then the nuisance nonparametric component is related to a nuisance parameter with increasing dimension in a purely parametric model. In the following I will Enno Mammen is Professor in Statistics, ). The author acknowledges support by the DFG project FOR916.give a more detailed discussion of this relation in the context of this article. DIMENSION ASYMPTOTICSHigh-dimensional models are a central example where asymptotic frameworks are used that do not neglect an important finite-sample feature in the asymptotic limit. Here, the important feature is the high dimensionality of the model. For high-dimensional models, this can be easily done by letting the dimension of the model grow with increasing sample size. Recently, there has been a huge amount of research on highdimensional models under sparsity constraints. This has also motivated investigators to revisit older strands of research a...

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Posterior inference in curved exponential families under increasing dimensions

Cited by 2 publications

References 28 publications

Generalized Jackknife Estimators of Weighted Average Derivatives

Generalized Jackknife Estimators of Weighted Average Derivatives

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