D-Trace precision matrix estimator with eigenvalue control

The estimation of a precision matrix has an important role in several research fields. In high-dimensional settings, one of the most prominent approaches to estimate the precision matrix is the ɭ₁ (Lasso) norm penalized convex optimization. This framework guarantees the sparsity of the estimated precision matrix. However, it does not control the eigenspectrum of the obtained estimator, and, moreover, it shrinks the largest eigenvalues of the estimated precision matrix. In this paper, we focus on D-trace precision matrix methodology. We propose imposing a negative trace penalization on the objective function of the D-trace approach, aimed to control the eigenvalues. Through extensive numerical analysis, using simulated and real datasets, we show the advantageous performance of our proposed methodology

New estimation methods for high dimensional inverse covariance matrices

The estimation of inverse covariance matrix (also known as precision matrix) is an important problem in various research fields and methodologies, especially in the current age of high-dimensional data abundance. In addition, the classical estimation methods are no longer stable and applicable in high dimensional settings, i.e., when the dimensionality has the same order as the sample size or is much larger. This thesis focuses on the estimation of the precision matrices as well as their applications. In particular, the goal of this thesis is to develop and analyse accurate precision matrix estimators for problems in high-dimensional settings. Moreover, the proposed precision matrix estimators should emulate the existing prominent estimators in terms of different statistical measures without being computationally more extensive. This thesis is comprised of two articles on estimation of precision matrices in high dimensional settings. In what follows, we summarize the main contributions of this thesis. First, we propose a simple improvement of the popular Graphical LASSO (GLASSO) framework that is able to attain better statistical performance without increasing signi cantly the computational cost. The proposed improvement is based on computing a root of the sample covariance matrix to reduce the spread of the associated eigenvalues. Through extensive numerical results, using both simulated and real datasets, we show that the proposed modiffication improves the GLASSO procedure. Our results reveal that the square-root improvement can be a reasonable choice in practice. Second, we introduce two adaptive extensions of the recently proposed l1 norm penalized D-trace loss minimization method. It is well known that the l1 norm penalization often fails to control the bias of the obtained estimator because of its overestimation behavior. Our proposed extensions are based on the adaptive and weighted adaptive thresholding operators and intend to diminish the bias produced by the l1 penalty term. We present the algorithm for solving our proposed approaches, which is based on the alternating direction method. Extensive numerical results, using both simulated and real datasets, show the advantage of our proposed estimators.Programa Oficial de Doctorado en Economía de la Empresa y Métodos CuantitativosPresidente: Francisco Javier Prieto Fernández; Secretario: Dae-Jin Lee Hwang; Vocal: Adolfo Hernández Estrad

Stable inverse probability weighting estimation for longitudinal studies

We consider estimation of the average effect of time-varying dichotomous exposure on outcome using inverse probability weighting (IPW) under the assumption that there is no unmeasured confounding of the exposure–outcome association at each time point. Despite the popularity of IPW, its performance is often poor due to instability of the estimated weights. We develop an estimating equation-based strategy for the nuisance parameters indexing the weights at each time point, aimed at preventing highly volatile weights and ensuring the stability of IPW estimation. Our proposed approach targets the estimation of the counterfactual mean under a chosen treatment regime and requires fitting a separate propensity score model at each time point. We discuss and examine extensions to enable the fitting of marginal structural models using one propensity score model across all time points. Extensive simulation studies demonstrate adequate performance of our approach compared with the maximum likelihood propensity score estimator and the covariate balancing propensity score estimator

D-trace Precision Matrix Estimation Using Adaptive Lasso Penalties

An accurate estimation of a precision matrix has a crucial role in the current age of high-dimensional data explosion. To deal with this problem, one of the prominent and commonly used techniques is the l1 norm (Lasso) penalization for a given loss function. This approach guarantees the sparsity of the precision matrix estimator for properly selected penalty parameters. However, the l1 norm penalization often fails to control the bias of the obtained estimator because of its overestimation behavior. In this paper, we introduce two adaptive extensions of the recently proposed l1 norm penalized D-trace loss minimization method. The proposed approaches intend to diminish the produced bias in the estimator. Extensive numerical results, using both simulated and real datasets, show the advantage of our proposed estimators

Improving the graphical lasso estimation for the precision matrix through roots ot the sample convariance matrix

In this paper, we focus on the estimation of a high-dimensional precision matrix. We propose a simple improvement of the graphical lasso framework (glasso) that is able to attain better statistical performance without sacrificing too much the computational cost. The proposed improvement is based on computing a root of the covariance matrix to reduce the spread of the associated eigenvalues, and maintains the original convergence rate. Through extensive numerical results, using both simulated and real datasets, we show the proposed modification outperforms the glasso procedure. Finally, our results show that the square-root improvement may be a reasonable choice in practiceAlonso gratefully acknowledge financial support from the Spanish Ministry of Science and Innovation grants ECO2011-25706 and ECO2012-3844

Doubly robust tests of exposure effects under high-dimensional confounding.

After variable selection, standard inferential procedures for regression parameters may not be uniformly valid; there is no finite-sample size at which a standard test is guaranteed to approximately attain its nominal size. This problem is exacerbated in high-dimensional settings, where variable selection becomes unavoidable. This has prompted a flurry of activity in developing uniformly valid hypothesis tests for a low-dimensional regression parameter (eg, the causal effect of an exposure A on an outcome Y) in high-dimensional models. So far there has been limited focus on model misspecification, although this is inevitable in high-dimensional settings. We propose tests of the null that are uniformly valid under sparsity conditions weaker than those typically invoked in the literature, assuming working models for the exposure and outcome are both correctly specified. When one of the models is misspecified, by amending the procedure for estimating the nuisance parameters, our tests continue to be valid; hence, they are doubly robust. Our proposals are straightforward to implement using existing software for penalized maximum likelihood estimation and do not require sample splitting. We illustrate them in simulations and an analysis of data obtained from the Ghent University intensive care unit

Improving the Graphical Lasso Estimation for the Precision Matrix Through Roots of the Sample Covariance Matrix

In this article, we focus on the estimation of a high-dimensional inverse covariance (i.e., precision) matrix. We propose a simple improvement of the graphical Lasso (glasso) framework that is able to attain better statistical performance without increasing significantly the computational cost. The proposed improvement is based on computing a root of the sample covariance matrix to reduce the spread of the associated eigenvalues. Through extensive numerical results, using both simulated and real datasets, we show that the proposed modification improves the glasso procedure. Our results reveal that the square-root improvement can be a reasonable choice in practice. Supplementary material for this article is available online.Andrés M. Alonso gratefully acknowledges financial support from CICYT Grants ECO2012-38442 and CO2015-66593. Francisco J. Nogales and Vahe Avagyan were supported by the Spanish Government through project MTM2013-44902-P

D-trace estimation of a precision matrix using adaptive Lasso penalties

The accurate estimation of a precision matrix plays a crucial role in the current age of high-dimensional data explosion. To deal with this problem, one of the prominent and commonly used techniques is the ℓ1 norm (Lasso) penalization for a given loss function. This approach guarantees the sparsity of the precision matrix estimate for properly selected penalty parameters. However, the ℓ1 norm penalization often fails to control the bias of obtained estimator because of its overestimation behavior. In this paper, we introduce two adaptive extensions of the recently proposed ℓ1 norm penalized D-trace loss minimization method. They aim at reducing the produced bias in the estimator. Extensive numerical results, using both simulated and real datasets, show the advantage of our proposed estimators.We would like to thank the Associate Editor, Coordinating Editor and two anonymous referees for their helpful comments that led to an improvement of this article. We express our gratitude to Teng Zhang and Hui Zou for sharing their Matlab code that solves the L1 norm penalized D-trace loss minimization problem. Andrés M. Alonso gratefully acknowledges financial support from CICYT (Spain) Grants ECO2012-38442 and ECO2015-66593. Francisco J. Nogales and Vahe Avagyan were supported by the Spanish Government through project MTM2013-44902-P. This paper is based on the first author's dissertation submitted to the Universidad Carlos III de Madrid. At the time of publication, Vahe Avagyan is a Postdoctoral fellow at Ghent University