Confidence sets based on penalized maximum likelihood estimators

Pötscher, Benedikt M. and Schneider, Ulrike (2008): Confidence sets based on penalized maximum likelihood estimators. Unpublished.

Abstract

The finite-sample coverage properties of confidence intervals based on penalized maximum likelihood estimators like the LASSO, adaptive LASSO, and hard-thresholding are analyzed. It is shown that symmetric intervals are the shortest. The length of the shortest intervals based on the hard-thresholding estimator is larger than the length of the shortest interval based on the adaptive LASSO, which is larger than the length of the shortest interval based on the LASSO, which in turn is larger than the standard interval based on the maximum likelihood estimator. In the case where the penalized estimators are tuned to possess the `sparsity property', the intervals based on these estimators are larger than the standard interval by an order of magnitude. A simple asymptotic confidence interval construction in the `sparse' case, that also applies to the smoothly clipped absolute deviation estimator, is also discussed.

• Item Type: MPRA Paper
• Language: English
• Keywords: penalized maximum likelihood, Lasso, adaptive Lasso, hard-thresholding, confidence set, coverage probability, sparsity, model selection.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods: General > C13 - Estimation; C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics

References

Fan, J. & R. Li (2001): Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348-1360.

Frank, I. E. & J. H. Friedman (1993): A statistical view of some chemometrics regression tools (with discussion). Technometrics 35, 109-148.

Joshi, V. M. (1969): Admissibility of the usual confidence sets for the mean of a univariate or bivariate normal population. Annals of Mathematical Statistics 40, 1042-1067.

Knight, K. & W. Fu (2000): Asymptotics of lasso-type estimators. Annals of Statistics 28, 1356-1378.

Leeb, H. & B. M. Pötscher (2008): Sparse estimators and the oracle property, or the return of Hodges' estimator. Journal of Econometrics 142, 201-211.

Pötscher, B. M. (2007): Confidence sets based on sparse estimators are necessarily large. Working Paper, Department of Statistics, University of Vienna. ArXiv: 0711.1036.

Pötscher, B. M. & H. Leeb (2007): On the distribution of penalized maximum likelihood estimators: the LASSO, SCAD, and thresholding. Working Paper, Department of Statistics, University of Vienna. ArXiv: 0711.0660.

Pötscher, B. M. & U. Schneider (2007): On the distribution of the adaptive LASSO estimator. Working Paper, Department of Statistics, University of Vienna. ArXiv: 0801.4627.

Tibshirani, R. (1996): Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B 58, 267-288.

Zou, H. (2006): The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101, 1418-1429.