Covariate Selection in High-Dimensional Generalized Linear Models With Measurement Error

In many problems involving generalized linear models, the covariates are subject to measurement error. When the number of covariates p exceeds the sample size n, regularized methods like the lasso or Dantzig selector are required. Several recent papers have studied methods which correct for measurement error in the lasso or Dantzig selector for linear models in the p > n setting. We study a correction for generalized linear models, based on Rosenbaum and Tsybakov’s matrix uncertainty selector. By not requiring an estimate of the measurement error covariance matrix, this generalized matrix uncertainty selector has a great practical advantage in problems involving high-dimensional data. We further derive an alternative method based on the lasso, and develop efficient algorithms for both methods. In our simulation studies of logistic and Poisson regression with measurement error, the proposed methods outperform the standard lasso and Dantzig selector with respect to covariate selection, by reducing the number of false positives considerably. We also consider classification of patients on the basis of gene expression data with noisy measurements. Supplementary materials for this article are available online.     

高维部分线性模型的变量选择和估计

考虑高维部分线性模型,提出了同时进行变量选择和估计兴趣参数的变量选择方法.将Dantzig变量选择应用到线性部分及非参数部分的各阶导数,从而获得参数和非参数部分的估计,且参数部分的估计具有稀疏性,证明了估计的非渐近理论界.最后,模拟研究了有限样本的性质.     

Analysis of definitive screening designs: Screening vs prediction

The use of definitive screening designs (DSDs) has been increasing since their introduction in 2011. These designs are used to screen factors and to make predictions. We assert that the choice of analysis method for these designs depends on the goal of the experiment, screening, or prediction. In this work, we present simulation results to address the explanatory (screening) use and the predictive use of DSDs. To address the predictive ability of DSDs, we use two 5‐factor DSDs and simultaneously run central composite designs case studies on which we will compare several common analysis methods. Overall, we find that for screening purposes, the Dantzig selector using the Bayesian Information Criterion statistic is a good analysis choice; however, when the goal of analysis is prediction forward selection using the Bayesian Information Criterion statistic produces models with a lower mean squared prediction error.     

Bayesian Object Identification: Variants

We present a Bayesian theory of object identification. Here, identifying an object means selecting a particular observation from a group of observations (variants), this observation (the regular variant) being characterized by a distributional model. In this sense, object identification means assigning a given model to one of several observations. Often, it is the statistical model of the regular variant, only, that is known. We study an estimator which relies essentially on this model and not on the characteristics of the “irregular” variants. In particular, we investigate under what conditions this variant selector is optimal. It turns out that there is a close relationship with exchangeability and Markovian reversibility. We finally apply our theory to the case of irregular variants generated from the regular variant by a Gaussian linear model.     

Graphical Criticism: Some Historical Notes

This article proposes a bootstrap local bandwidth selector for estimating nonparametric additive models. The selector is derived from a bootstrap approximation of the conditional mean squared error, based on a wild bootstrap resampling scheme applied to the estimated residuals. The selector is computed exactly (without involving Monte Carlo approximations) and in practice can be evaluated for many additive estimation methods, including backfitting (bivariate), marginal integration, and mixed methods. We study the consistency of the bootstrap approximation and also carry out an empirical simulation study to explore the performance of the proposed selector in comparison with others. The graphical tool SiZer Map enables us to make meaningful comparisons between local and global selectors.     

Bandwidth Selection in Density Estimation with Truncated and Censored Data

A plug-in type bandwidth selector is presented for density estimation with truncated and censored data. It is based on a representation of the MISE function obtained in the paper. Rate of convergence and limit distribution are derived for this selector. A bootstrap method is introduced to estimate the MISE whose minimizer is an alternative bandwidth selector. A simulation study was carried out to assess the behavior with small samples. This methodology is applied to a real-data problem consisting of reporting delay of AIDS cases. The almost sure representation of the product-limit estimator is a key tool in our proofs.     

Continuous selector of representative measures and the space of faces of a convex compactum

A sufficient condition for the existence of a continuous selector of representative measure, concentrated at the extreme points of a convex metrizable compactum, is considered. A necessary condition for the existence of such a selector is deduced. An example is given of a convex compactum with a closed set of extreme points, for which no continuous selector exists.Translated from Matematicheskie Zametki, Vol. 22, No. 6, pp. 897–906, December, 1977.In conclusion, the author expresses deep gratitude to A. M. Vershik for the formulation of the problem and for help in the preparation of this paper.     

Using bimodal kernel for inference in nonparametric regression with correlated errors

For nonparametric regression model with fixed design, it is well known that obtaining a correct bandwidth is difficult when errors are correlated. Various methods of bandwidth selection have been proposed, but their successful implementation critically depends on a tuning procedure which requires accurate information about error correlation. Unfortunately, such information is usually hard to obtain since errors are not observable. In this article a new bandwidth selector based on the use of a bimodal kernel is proposed and investigated. It is shown that the new bandwidth selector is quite useful for the tuning procedures of various other methods. Furthermore, the proposed bandwidth selector itself proves to be quite effective when the errors are severely correlated.     

The Dantzig selector for a linear model of diffusion processes

In this paper, a linear model of diffusion processes with unknown drift and diagonal diffusion matrices is discussed. We will consider the estimation problems for unknown parameters based on the discrete time observation in high-dimensional and sparse settings. To estimate drift matrices, the Dantzig selector which was proposed by Candés and Tao in 2007 will be applied. We will prove two types of consistency of the Dantzig selector for the drift matrix; one is the consistency in the sense of \(l_q\) norm for every \(q \in [1,\infty ]\) and another is the variable selection consistency. Moreover, we will construct an asymptotically normal estimator for the drift matrix by using the variable selection consistency of the Dantzig selector.

     

-additive families and the invariance of Borel classes

We prove that any -additive family of sets in an absolutely Souslin metric space has a -discrete refinement provided every partial selector set for is -discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps -sets to -sets and has complete fibers, admits a section of the first class. The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves -sets, is shown as an application of the previous result.

     

An effective bandwidth selector in a complicated kernel regression

The field of nonparametrics has shown its appeal in recent years with an array of new tools for statistical analysis. As one of those tools, nonparametric regression has become a prominent statistical research topic and also has been well established as a useful tool. In this article we investigate the biased cross-validation selector, BCV, which is proposed by Ohet al. (1995), for a less smoothing regression function. In the simulation study, BCV selector is shown to perform well in practice with respect to ASE ratio.     

An application of the Lefschetz fixed-point theorem to non-convex differential inclusions on manifolds

A selector theorem for non-convex orientor fields on closed manifolds is given and the Lefschetz fixed point theorem is used to establish an existence result for these ones.

     

Looking for Selectors of Star Bodies

Let be a family of star bodies in R ⁿ (compact bodies in R ⁿ with nonempty kernels). A function s: R ⁿ is a selector for provided that s(A) ker A for every A . To every star body A and every : [0,) [0,) we assign a function _A: ker A R defined in terms of the radial function of translates of A. We prove that if A is convex and is concave and strictly increasing, then _A has a unique maximizer, which is referred to as the radial center of A induced by (Theorem 3.1). We extend the radial center map over some family of star bodies (Theorem 4.2). Further, we define a suitable metric, _st , the star metric, on the family of all the compact star sets in R ⁿ . This new metric is topologically stronger than the Hausdorff metric (Theorem 5.7). We study the continuity of our selectors with respect to _st .     

A selector theorem in Banach spaces

In the present paper, we prove a selector theorem in Banach spaces similar to that of Filippov (Ref. 1), but where the measurability of the selected function follows from a weak upper semicontinuity property.     

Existence and relaxation theorems for extreme continuous selectors of multifunctions with decomposable values

We present some extreme continuous selector theorems, synthesizing the author's results; namely, we study existence and properties of continuous selectors from the set of extreme points of multifunctions with closed convex decomposable values in the space of Bochner integrable functions.     

On the Equivalence of Borel Sets

Mathematical Notes - A theorem on the existence of a definable selector for the relation $$E_{\aleph_0}$$ on the $$\boldsymbol{\Delta}^0_2$$ -sets of the Baire space $$\omega^\omega$$ is proved.     

Optimal stopping in a search for a vertex with full degree in a random graph

We consider the following on-line decision problem. The vertices of a realization of the random graph G(n,p) are being observed one by one by a selector. At time m, the selector examines the mth vertex and knows the graph induced by the m vertices that have already been examined. The selector’s aim is to choose the currently examined vertex maximizing the probability that this vertex has full degree, i.e. it is connected to all other vertices in the graph. An optimal algorithm for such a choice (in other words, optimal stopping time) is given. We show that it is of a threshold type and we find the threshold and its asymptotic estimation.     

发散维数SICA惩罚Cox回归模型的一种修正BIC调节参数选择器

本文研究了发散维数SICA惩罚Cox回归模型的调节参数选择问题,提出了一种修正的BIC调节参数选择器.在一定的正则条件下,证明了方法的模型选择相合性.数值结果表明提出的方法表现要优于GCV准则.     

Selectors in non-Archimedean spaces

Continuous selectors on the hyperspace F(X) are studied, when X is a non-Archimedean space. It is shown that a non-Archimedean space has a continuous selector if and only if it is topologically well orderable. Another characterization is given in terms of density and complete metrizability.