arXiv 19 Dec 2015 · Mathematics — Statistics Theory · 3 citations (OpenAlex)
arXiv:1512.06290 · PDF · DOI · OpenAlex · Extracted main text
We propose a general framework for regularization in M-estimation problems under time dependent (absolutely regular-mixing) data which encompasses many of the existing estimators. We derive non-asymptotic concentration bounds for the regularized M-estimator. Our results exhibit a variance-bias trade-off, with the variance term being governed by a novel measure of the complexity of the parameter set. We also show that the mixing structure affect the variance term by scaling the number of observations; depending on the decay rate of the mixing coefficients, this scaling can even affect the asymptotic behavior. Finally, we propose a data-driven method for choosing the tuning parameters of the regularized estimator which yield the same (up to constants) concentration bound as one that optimally balances the (squared) bias and variance terms. We illustrate the results with several canonical examples.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | X. Chen and X. Shen (1998) Sieve extremum estimates for weakly dependent data | 1.000 | 8 | 4 | 100% |
| 2 | S. Chatterjee (2013) Assumptionless consistency of the Lasso | 1.000 | 7 | 3 | 100% |
| 3 | P. Bickel and B. Li (2006) Regularization in statistics | 1.000 | 5 | 3 | 100% |
| 4 | P. Doukhan, P. Massart, and E. Rio (1995) Invariance principles for absolutely regular empirical processes | 0.941 | 6 | 5 | 83% |
| 5 | S. van de Geer and P. Bülmann (2011) Statistics for High-Dimensional Data | 0.941 | 6 | 4 | 83% |
| 6 | M. Talagrand (2014) Upper and Lower Bounds for Stochastic Processes | 0.909 | 16 | 6 | 75% |
| 7 | S. Pereverzev and E. Schock (2006) On the adaptive selection of the parameter in regularization of ill-posed problems | 0.811 | 4 | 2 | 100% |
| 8 | A.W. van der Vaart and J. Wellner (1996) Weak Convergence and Empirical Processes with Applications to Statistics | 0.737 | 5 | 4 | 40% |
| 9 | X. Chen and D. Pouzo (2012) Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals | 0.737 | 4 | 4 | 50% |
| 10 | X. Chen (2013) Penalized sieve estimation and inference of seminonparametric dynamic models: A selective review | 0.737 | 3 | 2 | 100% |
Showing the top 10 of 66 scored citations.