Detangling robustness in high dimensions: Composite versus model-averaged estimation

Zhou, Jing; Claeskens, Gerda; Bradić, Jelena

doi:10.1214/20-ejs1728

Cited by 4 publications

(16 citation statements)

References 26 publications

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“…This is in contrast with so‐called oracle approaches that assume perfect selection. Further, the mean squared difference of the AMP approximation and

\hat{\beta}\left(\lambda \right)

converges to 0 almost surely (Bayati & Montanari, 2011b, theorem 1.8) when

p\to \infty

and similar results are obtained for other loss functions (Bradic, 2016; Zhou et al, 2020).…”

Section: Notation Assumptions and Estimatorssupporting

confidence: 65%

“…We assume the design matrix

X\in {\mathbb{R}}^{n\times p}

, the error vector

\boldsymbol{\varepsilon}

, and the coefficient vector

\beta

to satisfy Assumptions (A1)–(A5) from Zhou et al (2020, appendix A), which we repeat here for completeness. A standard Gaussian design: for

i=1,\dots, n

and

j=1,\dots, p

, the

{X}_{ij}\sim N\left(0,1/n\right)

are independent and identically distributed. For the

p

‐vector

\beta

it holds that for

p

tending to infinity a sequence of uniform distributions that is placed on its components converges to a distribution with a bounded

\left(2k-2\right)

th moment for

k\ge 2

.…”

Section: Notation Assumptions and Estimatorsmentioning

confidence: 99%

“…with cumulative distribution function

{F}_{\varepsilon }

and density function

{f}_{\varepsilon }

, which is used to determine the intervals in Assumption (A3). Assumption (A4) has been used in Bayati and Montanari (2011a) and Zhou et al (2020, lemma 1) there taking

\kappa =2

.…”

Section: Notation Assumptions and Estimatorsmentioning

confidence: 99%

“…The RAMP algorithm (see Bradic, 2016; Huang, 2020); Zhou et al (2020, section 3.2) is an iterative procedure consisting of three steps. The iteration number is denoted by

t=1,2,\dots

.…”

Section: Notation Assumptions and Estimatorsmentioning

confidence: 99%

“…Instead, our method relies on an asymptotically normal distributed estimator obtained from the robust approximate message passing (RAMP) algorithm (Bayati & Montanari, 2011a; Bradic, 2016; Donoho, Maleki, & Montanari, 2009; Donoho & Montanari, 2016; Zhou, Claeskens, & Bradic, 2020). The estimator in the last step of the RAMP algorithm is obtained by applying a soft‐thresholding function to the estimator that is our main focus.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Automatic bias correction for testing in high‐dimensional linear models

Zhou

Claeskens

2022

Statistica Neerlandica

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Hypothesis testing is challenging due to the test statistic's complicated asymptotic distribution when it is based on a regularized estimator in high dimensions. We propose a robust testing framework for 𝓁 1 -regularized M-estimators to cope with non-Gaussian distributed regression errors, using the robust approximate message passing algorithm. The proposed framework enjoys an automatically built-in bias correction and is applicable with general convex nondifferentiable loss functions which also allows inference when the focus is a conditional quantile instead of the mean of the response. The estimator compares numerically well with the debiased and desparsified approaches while using the least squares loss function. The use of the Huber loss function demonstrates that the proposed construction provides stable confidence intervals under different regression error distributions.

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“…This is in contrast with so‐called oracle approaches that assume perfect selection. Further, the mean squared difference of the AMP approximation and

\hat{\beta}\left(\lambda \right)

converges to 0 almost surely (Bayati & Montanari, 2011b, theorem 1.8) when

p\to \infty

and similar results are obtained for other loss functions (Bradic, 2016; Zhou et al, 2020).…”

Section: Notation Assumptions and Estimatorssupporting

confidence: 65%

“…We assume the design matrix

X\in {\mathbb{R}}^{n\times p}

, the error vector

\boldsymbol{\varepsilon}

, and the coefficient vector

\beta

to satisfy Assumptions (A1)–(A5) from Zhou et al (2020, appendix A), which we repeat here for completeness. A standard Gaussian design: for

i=1,\dots, n

and

j=1,\dots, p

, the

{X}_{ij}\sim N\left(0,1/n\right)

are independent and identically distributed. For the

p

‐vector

\beta

it holds that for

p

tending to infinity a sequence of uniform distributions that is placed on its components converges to a distribution with a bounded

\left(2k-2\right)

th moment for

k\ge 2

.…”

Section: Notation Assumptions and Estimatorsmentioning

confidence: 99%

“…with cumulative distribution function

{F}_{\varepsilon }

and density function

{f}_{\varepsilon }

, which is used to determine the intervals in Assumption (A3). Assumption (A4) has been used in Bayati and Montanari (2011a) and Zhou et al (2020, lemma 1) there taking

\kappa =2

.…”

Section: Notation Assumptions and Estimatorsmentioning

confidence: 99%

“…The RAMP algorithm (see Bradic, 2016; Huang, 2020); Zhou et al (2020, section 3.2) is an iterative procedure consisting of three steps. The iteration number is denoted by