Rates of convergence of the Adaptive LASSO estimators to the Oracle distribution and higher order refinements by the bootstrap

Chatterjee, Arindam; Lahiri, Soumendra N.

doi:10.1214/13-aos1106

Cited by 86 publications

(72 citation statements)

References 29 publications

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“…In other words, we have

ϱ false(H^{b}, H false) \overset{p}{\to} 0

where

ϱ false(\cdot, \cdot false)

represents the Prohorov metric on the set of all probability measures on

false({double-struckℝ}^{p}, scriptB false({double-struckℝ}^{p} false) false)

, and

H^{b}

and H are the asymptotic distributions of the centered and scaled estimates,

\sqrt{n} false(\overset{boldθ}{true} - θ false)

and

\sqrt{n} false({\overset{trueˆ}{θ}}^{false(b false)} - \overset{boldθ}{true} false)

. Similarly, residual bootstrap is able to produce a valid sampling distribution for SCAD, MCP and other studentized estimators Chatterjee and Lahiri (). Throughout the article, we mostly adopt residual bootstrap.…”

Section: Methodsmentioning

confidence: 99%

Model Confidence Bounds for Variable Selection

Luo

Ferrari

et al. 2019

Biometrics

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In this article, we introduce the concept of model confidence bounds (MCB) for variable selection in the context of nested models. Similarly to the endpoints in the familiar confidence interval for parameter estimation, the MCB identifies two nested models (upper and lower confidence bound models) containing the true model at a given level of confidence. Instead of trusting a single selected model obtained from a given model selection method, the MCB proposes a group of nested models as candidates and the MCB's width and composition enable the practitioner to assess the overall model selection uncertainty. A new graphical tool—the model uncertainty curve (MUC)—is introduced to visualize the variability of model selection and to compare different model selection procedures. The MCB methodology is implemented by a fast bootstrap algorithm that is shown to yield the correct asymptotic coverage under rather general conditions. Our Monte Carlo simulations and real data examples confirm the validity and illustrate the advantages of the proposed method.

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“…In other words, we have

ϱ false(H^{b}, H false) \overset{p}{\to} 0

where

ϱ false(\cdot, \cdot false)

represents the Prohorov metric on the set of all probability measures on

false({double-struckℝ}^{p}, scriptB false({double-struckℝ}^{p} false) false)

, and

H^{b}

and H are the asymptotic distributions of the centered and scaled estimates,

\sqrt{n} false(\overset{boldθ}{true} - θ false)

and

\sqrt{n} false({\overset{trueˆ}{θ}}^{false(b false)} - \overset{boldθ}{true} false)

Section: Methodsmentioning

confidence: 99%

Model Confidence Bounds for Variable Selection

Luo

Ferrari

et al. 2019

Biometrics

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show abstract

“…Recently, many studies have analyzed the validity of resampling methods for lasso estimators in i.i.d. settings; see, e.g., Chatterjee and Lahiri, , Minnier et al , Efron , and Camponovo . In this paper, we extend the bootstrap procedure introduced in Camponovo to more general time series models, and prove its (pointwise) consistency.…”

Section: Introductionmentioning

confidence: 86%

Oracle Properties, Bias Correction, and Bootstrap Inference for Adaptive Lasso for Time Series M‐Estimators

Audrino

Camponovo

2017

Journal Time Series Analysis

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We derive new theoretical results on the properties of the adaptive least absolute shrinkage and selection operator (adaptive lasso) for possibly nonlinear time series models. In particular, we investigate the question of how to conduct inference on the parameters given an adaptive lasso model. Central to this study is the test of the hypothesis that a given adaptive lasso parameter equals zero, which therefore tests for a false positive. To this end, we introduce a recentered bootstrap procedure and show, theoretically and empirically through extensive Monte Carlo simulations, that the adaptive lasso can combine efficient parameter estimation, variable selection, and inference in one step. Moreover, we analytically derive a bias correction factor that is able to significantly improve the empirical coverage of the test on the active variables. Finally, we apply the adaptive lasso and the recentered bootstrap procedure to investigate the relation between the short rate dynamics and the economy, thereby providing a statistical foundation (from a model choice perspective) for the classic Taylor rule monetary policy model.

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“…Providing uniform asymptotic results under the varying‐parameter setting is indeed important for constructing a hypothesis test or confidence interval when focusing on the magnitude of the parameters (Dezeure et al., ), but the fixed‐parameter setting is still an important case that requires thorough investigation. Recent research has also been conducted on the fixed‐parameter setting, such as (Chatterjee and Lahiri, ; Chatterjee and Lahiri, ) and (Liu and Yu, ). Our article focuses on the fixed parameter setting with asymptotic properties established by letting the sample size go to infinity.…”

Section: Leeb Pötscher and Kivaranovicmentioning

confidence: 99%