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Register a new userHigher Order Refinements by Bootstrap in Lasso and other Penalized Regression Methods
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Added by
Debraj Das
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
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Selection of important covariates and to drop the unimportant ones from a high-dimensional regression model is a long standing problem and hence have received lots of attention in the last two decades. After selecting the correct model, it is also important to properly estimate the existing parameters corresponding to important covariates. In this spirit, Fan and Li (2001) proposed Oracle property as a desired feature of a variable selection method. Oracle property has two parts; one is the variable selection consistency (VSC) and the other one is the asymptotic normality. Keeping VSC fixed and making the other part stronger, Fan and Lv (2008) introduced the strong oracle property. In this paper, we consider different penalized regression techniques which are VSC and classify those based on oracle and strong oracle property. We show that both the residual and the perturbation bootstrap methods are second order correct for any penalized estimator irrespective of its class. Most interesting of all is the Lasso, introduced by Tibshirani (1996). Although Lasso is VSC, it is not asymptotically normal and hence fails to satisfy the oracle property.
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In the fields of clinical trials, biomedical surveys, marketing, banking, with dichotomous response variable, the logistic regression is considered as an alternative convenient approach to linear regression. In this paper, we develop a novel bootstrap technique based on perturbation resampling method for approximating the distribution of the maximum likelihood estimator (MLE) of the regression parameter vector. We establish second order correctness of the proposed bootstrap method after proper studentization and smoothing. It is shown that inferences drawn based on the proposed bootstrap method are more accurate compared to that based on asymptotic normality. The main challenge in establishing second order correctness remains in the fact that the response variable being binary, the resulting MLE has a lattice structure. We show the direct bootstrapping approach fails even after studentization. We adopt smoothing technique developed in Lahiri (1993) to ensure that the smoothed studentized version of the MLE has a density. Similar smoothing strategy is employed to the bootstrap version also to achieve second order correct approximation.
In this paper, we develop uniform inference methods for the conditional mode based on quantile regression. Specifically, we propose to estimate the conditional mode by minimizing the derivative of the estimated conditional quantile function defined by smoothing the linear quantile regression estimator, and develop two bootstrap methods, a novel pivotal bootstrap and the nonparametric bootstrap, for our conditional mode estimator. Building on high-dimensional Gaussian approximation techniques, we establish the validity of simultaneous confidence rectangles constructed from the two bootstrap methods for the conditional mode. We also extend the preceding analysis to the case where the dimension of the covariate vector is increasing with the sample size. Finally, we conduct simulation experiments and a real data analysis using U.S. wage data to demonstrate the finite sample performance of our inference method.
From an optimizers perspective, achieving the global optimum for a general nonconvex problem is often provably NP-hard using the classical worst-case analysis. In the case of Coxs proportional hazards model, by taking its statistical model structures into account, we identify local strong convexity near the global optimum, motivated by which we propose to use two convex programs to optimize the folded-concave penalized Coxs proportional hazards regression. Theoretically, we investigate the statistical and computational tradeoffs of the proposed algorithm and establish the strong oracle property of the resulting estimators. Numerical studies and real data analysis lend further support to our algorithm and theory.
We generalize standard credal set models for imprecise probabilities to include higher order credal sets -- confidences about confidences. In doing so, we specify how an agents higher order confidences (credal sets) update upon observing an event. Our model begins to address standard issues with imprecise probability models, like Dilation and Belief Inertia. We conjecture that when higher order credal sets contain all possible probability functions, then in the limiting case the highest order confidences converge to form a uniform distribution over the first order credal set, where we define uniformity in terms of the statistical distance metric (total variation distance). Finite simulation supports the conjecture. We further suggest that this convergence presents the total-variation-uniform distribution as a natural, privileged prior for statistical hypothesis testing.
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.
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