Confidence sets based on penalized maximum likelihood estimators in Gaussian regression

By B. M. Potscher, Ulrike Schneider. 2008

53 citations recorded by the source corpus.

Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and 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. Furthermore, a simple asymptotic confidence interval construction in the `sparse' case, that also applies to the smoothly clipped absolute deviation estimator, is discussed. The results for the known-variance case are shown to carry over to the unknown-variance case in an appropriate asymptotic sense.

Subfields: Control and Systems Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3352e59ce4a58e9f12e0942769c1fbd57319d5f2

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In Control and Systems Engineering

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Citations in the corpus

On various confidence intervals post-model-selection (2014, unknown): “…(1963); Dijkstra and Veldkamp (1988); Kabaila (1998, 2009); Kabaila and Leeb (2006); Leeb (2006); Leeb and P¨otscher (2003, 2005, 2006a,b, 2008a,b); Olshen (1973); P¨otscher (1991, 2006); P¨otscher and Leeb (2009); P¨otscher and Schneider (2009, 2010, 2011); Sen (1979); Sen and Saleh (1987).”
Bonferroni-Based Size-Correction for Nonstandard Testing Problems (2017, unknown): “Though some very recent work has explored the properties of uniformly valid confidence intervals for some of the problems that fall into this framework (e.g., Pötscher and Schneider, 2010), to the author’s knowledge, this is the first time such a framework and corresponding uniformly valid…”
Valid post-selection inference (2013, background): “…by the “Vienna School” of Pötscher, Leeb and Schneider; see, for example, Leeb and Pötscher (2003, 2005, 2006a, 2006b, 2008a, 2008b, 2008c), Pötscher (2006), Leeb (2006), Pötscher and Leeb (2009), Pötscher and Schneider (2009, 2010, 2011), as well as Kabaila and Leeb (2006) and Kabaila (2009).”
Model confidence bounds for variable selection (2016, result): “Pötscher and Schneider (2010) and Leeb and Pötscher (2006) have presented similar assumptions in their results.”