High-dimensional covariance matrix estimation using a low-rank and diagonal decomposition

We study high-dimensional covariance/precision matrix estimation under the assumption that the covariance/precision matrix can be decomposed into a low-rank component $L$ and a diagonal component $D$. The rank of $L$ can either be chosen to be small or controlled by a penalty function. Under moderate conditions on the population covariance/precision matrix itself and on the penalty function, we prove some consistency results for our estimators. A block-wise coordinate descent algorithm, which iteratively updates $L$ and $D$, is then proposed to obtain the estimator in practice. Finally, various numerical experiments are presented; using simulated data, we show that our estimator performs quite well in terms of the Kullback–Leibler loss; using stock return data, we show that our method can be applied to obtain enhanced solutions to the Markowitz portfolio selection problem. The Canadian Journal of Statistics 48: 308–337; 2020 © 2019 Statistical Society of Canada     

Direct estimation of differential networks under high-dimensional nonparanormal graphical models

In genomics, it is often of interest to study the structural change of a genetic network between two phenotypes. Under Gaussian graphical models, the problem can be transformed to estimating the difference between two precision matrices, and several approaches have been recently developed for this task such as joint graphical lasso and fused graphical lasso. However, the multivariate Gaussian assumptions made in the existing approaches are often violated in reality. For instance, most RNA-Seq data follow non-Gaussian distributions even after log-transformation or other variance-stabilizing transformations. In this work, we consider the problem of directly estimating differential networks under a flexible semiparametric model, namely the nonparanormal graphical model, where the random variables are assumed to follow a multivariate Gaussian distribution after a set of monotonically increasing transformations. We propose to use a novel rank-based estimator to directly estimate the differential network, together with a parametric simplex algorithm for fast implementation. Theoretical properties of the new estimator are established under a high-dimensional setting where $p$ grows with $n$ almost exponentially fast. In particular, we show that the proposed estimator is consistent in both parameter estimation and support recovery. Both synthetic data and real genomic data are used to illustrate the promise of the new approach. The Canadian Journal of Statistics 48: 187–203; 2020 © 2019 Statistical Society of Canada     

Optimal nonparametric estimator of the area under ROC curve based on clustered data

Abstract

In diagnostic trials, clustered data are obtained when several subunits of the same patient are observed. Intracluster correlations need to be taken into account when analyzing such clustered data. A nonparametric method has been proposed by Obuchowski (1997 Obuchowski, N. A. 1997. Nonparametric analysis of clustered ROC curve data. Biometrics 53 (2):567–78.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to estimate the Receiver Operating Characteristic curve area (AUC) for such clustered data. However, Obuchowski’s estimator is not efficient as it gives equal weight to all pairwise rankings within and between cluster. In this paper, we propose a more efficient nonparametric AUC estimator with two sets of optimal weights. Simulation results show that the loss of efficiency of Obuchowski’s estimator for a single AUC or the AUC difference can be substantial when there is a moderate intracluster test correlation and the cluster size is large. The efficiency gain of our weighted AUC estimator for a single AUC or the AUC difference is further illustrated using the data from a study of screening tests for neonatal hearing.     

Test for comparing complete expectations of life of two groups

Abstract

Complete expectation of life of an individual gives an intuitive and interesting perspective on the ageing process and is an important concept in the insurance sector for determination of premium. We propose a new test for testing equality of complete expectations of life of two groups/populations. Power of the new test is calculated through simulations and compared with the power of the tests given by Berger, Boos, and Guess (1988 Berger, R. L., D. D. Boos, and F. M. Guess. 1988. Tests and confidence sets for comparing two mean residual life functions. Biometrics 44 (1):103–15.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Aly (1997 Aly, E. E. A. A. 1997. Nonparametric tests for comparing two mean residual life functions. Lifetime Data Analysis 3 (4):353–66.[Crossref], [PubMed] , [Google Scholar]). It is observed that the proposed test statistic is more powerful than the competing tests for the cases considered in this paper. A real life illustration is included.     

Hypothesis testing via a penalized-likelihood approach

It is illustrated in this paper that hypothesis testing procedures can be derived based on the penalized likelihood approach. Based on this point of view, many traditional hypothesis tests, including the two-sample mean test, score test, and Hotelling’s $T^2$ test are revisited under the penalized likelihood framework. Similar framework is also applicable to the empirical likelihood.     

Stepwise Regression in Mixed Quantitative Linear Models with Autocorrelated Errors

ABSTRACT

In the stepwise procedure of selection of a fixed or a random explanatory variable in a mixed quantitative linear model with errors following a Gaussian stationary autocorrelated process, we have studied the efficiency of five estimators relative to Generalized Least Squares (GLS): Ordinary Least Squares (OLS), Maximum Likelihood (ML), Restricted Maximum Likelihood (REML), First Differences (FD), and First-Difference Ratios (FDR). We have also studied the validity and power of seven derived testing procedures, to assess the significance of the slope of the candidate explanatory variable x ₂ to enter the model in which there is already one regressor x ₁. In addition to five testing procedures of the literature, we considered the FDR t-test with n ? 3 df and the modified t-test with n? ? 3 df for partial correlations, where n? is Dutilleul's effective sample size. Efficiency, validity, and power were analyzed by Monte Carlo simulations, as functions of the nature, fixed vs. random (purely random or autocorrelated), of x ₁ and x ₂, the sample size and the autocorrelation of random terms in the regression model. We report extensive results for the autocorrelation structure of first-order autoregressive [AR(1)] type, and discuss results we obtained for other autocorrelation structures, such as spherical semivariogram, first-order moving average [MA(1)] and ARMA(1,1), but we could not present because of space constraints. Overall, we found that:

1. the efficiency of slope estimators and the validity of testing procedures depend primarily on the nature of x ₂, but not on that of x ₁;

2. FDR is the most inefficient slope estimator, regardless of the nature of x ₁ and x ₂;

3. REML is the most efficient of the slope estimators compared relative to GLS, provided the specified autocorrelation structure is correct and the sample size is large enough to ensure the convergence of its optimization algorithm;

4. the FDR t-test, the modified t-test and the REML t-test are the most valid of the testing procedures compared, despite the inefficiency of the FDR and OLS slope estimators for the former two;

5. the FDR t-test, however, suffers from a lack of power that varies with the nature of x ₁ and x ₂; and

6. the modified t-test for partial correlations, which does not require the specification of an autocorrelation structure, can be recommended when x ₁ is fixed or random and x ₂ is random, whether purely random or autocorrelated. Our results are illustrated by the environmental data that motivated our work.