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The lasso is a popular technique for simultaneous estimation and variable selection. Lasso variable selection has been shown to be consistent under certain conditions. In this work we derive a necessary condition for the lasso variable selection to be consistent. Consequently, there exist certain scenarios where the lasso is inconsistent for variable selection. We then propose a new version of the lasso, called the adaptive lasso, where adaptive weights are used for penalizing different coefficients in the ℓ 1 penalty. We show that the adaptive lasso enjoys the oracle properties; namely, it performs as well as if the true underlying model were given in advance. Similar to the lasso, the adaptive lasso is shown to be near-minimax optimal. Furthermore, the adaptive lasso can be solved by the same efficient algorithm for solving the lasso. We also discuss the extension of the adaptive lasso in generalized linear models and show that the oracle properties still hold under mild regularity conditions. As a byproduct of our theory, the nonnegative garotte is shown to be consistent for variable selection.

Kriging models are popular in analyzing computer experiments. The most widely used kriging models apply a constant mean to capture the overall trend. This method can lead to a poor prediction when strong trends exist. To tackle this problem, a new modeling method is proposed, which incorporates a variable selection mechanism into kriging via a penalty function. An efficient algorithm is introduced and oracle properties in terms of selecting the correct mean function are derived according to fixed-domain asymptotics. The finite-sample performance is examined via a simulation study. Application of the proposed methodology to circuit-simulation experiments demonstrates a remarkable improvement in prediction, and the capability of identifying variables that most affect the system.

Estimation of the number or proportion of true null hypotheses in multiple-testing problems has become an interesting area of research. The first important work in this field was performed by Schweder and Spjøtvoll. Among others, they proposed to use plug-in estimates for the proportion of true null hypotheses in multiple-test procedures to improve the power. We investigate the problem of controlling the familywise error rate FWER when such estimators are used as plug-in estimators in single-step or step-down multiple-test procedures. First we investigate the case of independent p-values under the null hypotheses and show that a suitable choice of plug-in estimates leads to control of FWER in single-step procedures. We also investigate the power and study the asymptotic behaviour of the number of false rejections. Although step-down procedures are more difficult to handle we briefly consider a possible solution to this problem. Anyhow, plug-in step-down procedures are not recommended here. For dependent p-values we derive a condition for asymptotic control of FWER and provide some simulations with respect to FWER and power for various models and hypotheses.

Variable selection is an important issue in all regression analyses, and in this paper we discuss this in the context of regression analysis of panel count data. Panel count data often occur in long-term studies that concern occurrence rate of a recurrent event, and their analysis has recently attracted a great deal of attention. However, there does not seem to exist any established approach for variable selection with respect to panel count data. For the problem, we adopt the idea behind the non-concave penalized likelihood approach and develop a non-concave penalized estimating function approach. The proposed methodology selects variables and estimates regression coefficients simultaneously, and an algorithm is presented for this process. We show that the proposed procedure performs as well as the oracle procedure in that it yields the estimates as if the correct submodel were known. Simulation studies are conducted for assessing the performance of the proposed approach and suggest that it works well for practical situations. An illustrative example from a cancer study is provided.

Variable selection is an important issue in all regression analysis and in this paper, we discuss this in the context of regression analysis of recurrent event data. Recurrent event data often occur in long-term studies in which individuals may experience the events of interest more than once and their analysis has recently attracted a great deal of attention (Andersen et al., Statistical models based on counting processes, 1993; Cook and Lawless, Biometrics 52:1311–1323, 1996, The analysis of recurrent event data, 2007; Cook et al., Biometrics 52:557–571, 1996; Lawless and Nadeau, Technometrics 37:158-168, 1995; Lin et al., J R Stat Soc B 69:711–730, 2000). However, it seems that there are no established approaches to the variable selection with respect to recurrent event data. For the problem, we adopt the idea behind the nonconcave penalized likelihood approach proposed in Fan and Li (J Am Stat Assoc 96:1348–1360, 2001) and develop a nonconcave penalized estimating function approach. The proposed approach selects variables and estimates regression coefficients simultaneously and an algorithm is presented for this process. We show that the proposed approach performs as well as the oracle procedure in that it yields the estimates as if the correct submodel was known. Simulation studies are conducted for assessing the performance of the proposed approach and suggest that it works well for practical situations. The proposed methodology is illustrated by using the data from a chronic granulomatous disease study.

The concept of k-FWER has received much attention lately as an appropriate error rate for multiple testing when one seeks to control at least k false rejections, for some fixed k ≥ 1. A less conservative notion, the k-FDR, has been introduced very recently by Sarkar [Ann. Statist. 34 (2006) 394-415], generalizing the false discovery rate of Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300]. In this article, we bring newer insight to the k-FDR considering a mixture model involving independent p-values before motivating the developments of some new procedures that control it. We prove the k-FDR control of the proposed methods under a slightly weaker condition than in the mixture model. We provide numerical evidence of the proposed methods' superior power performance over some k-FWER and k-FDR methods. Finally, we apply our methods to a real data set.

Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization algorithm possesses those nice theoretical properties. To close this important theoretical gap in over a decade, we provide a unified theory to show explicitly how to obtain the oracle solution via the local linear approximation algorithm. For a folded concave penalized estimation problem, we show that as long as the problem is localizable and the oracle estimator is well behaved, we can obtain the oracle estimator by using the one-step local linear approximation. In addition, once the oracle estimator is obtained, the local linear approximation algorithm converges, namely it produces the same estimator in the next iteration. The general theory is demonstrated by using four classical sparse estimation problems, that is, sparse linear regression, sparse logistic regression, sparse precision matrix estimation and sparse quantile regression.

We propose dimension reduction methods for sparse, high-dimensional multivariate response regression models. Both the number of responses and that of the predictors may exceed the sample size. Sometimes viewed as complementary, predictor selection and rank reduction are the most popular strategies for obtaining lower-dimensional approximations of the parameter matrix in such models. We show in this article that important gains in prediction accuracy can be obtained by considering them jointly. We motivate a new class of sparse multivariate regression models, in which the coefficient matrix has low rank and zero rows or can be well approximated by such a matrix. Next, we introduce estimators that are based on penalized least squares, with novel penalties that impose simultaneous row and rank restrictions on the coefficient matrix. We prove that these estimators indeed adapt to the unknown matrix sparsity and have fast rates of convergence. We support our theoretical results with an extensive simulation study and two data analyses.

Standard assumptions incorporated into Bayesian model selection procedures result in procedures that are not competitive with commonly used penalized likelihood methods. We propose modifications of these methods by imposing nonlocal prior densities on model parameters. We show that the resulting model selection procedures are consistent in linear model settings when the number of possible covariates p is bounded by the number of observations n, a property that has not been extended to other model selection procedures. In addition to consistently identifying the true model, the proposed procedures provide accurate estimates of the posterior probability that each identified model is correct. Through simulation studies, we demonstrate that these model selection procedures perform as well or better than commonly used penalized likelihood methods in a range of simulation settings. Proofs of the primary theorems are provided in the Supplementary Material that is available online.

Decentralized Finance (DeFi) takes the promise of blockchain a step further and aims to transform traditional financial products into trustless and transparent protocols that run without involving intermediaries. Similar to how 2017 was the year of ICOs, 2020 was the year of DeFi, with more than fifteen billion dollars of total investments. The decentralized platforms utilize oracles to retrieve asset data from the external world, but their choice and management criteria are often unknown to the end-users. If oracles are poorly selected or managed, the funds of a rising number of investors are inevitably in danger. The issue, known as “the oracle problem”, which makes real-world applications controversial and debated due to the loss of decentralization, had recently drawn attention to DeFi, given the crescent number of related hacks that caused the loss of millions of dollars held in DeFi projects. Through a multivocal approach that considers academic papers, whitepapers, preprints, and opinion posts, this study aims to shed light on the pattern that identifies the oracle problem in DeFi and outline the most promising ways to overcome the related weaknesses. This research supports the view that the oracle problem in decentralized finance bears specific characteristics which require standardization and appropriate economic incentives to be addressed.