arXiv subjects

Lee H. Dicker

Publications and source records attributed to Lee H. Dicker.

8 recordsLinked to original sources

arXiv2019

The Mahalanobis kernel for heritability estimation in genome-wide association studies: fixed-effects and random-effects methods

Linear mixed models (LMMs) are widely used for heritability estimation in genome-wide association studies (GWAS). In standard approaches to heritability estimation with LMMs, a genetic relationship matrix (GRM) must be specified. In GWAS, the GRM is frequently a correlation matrix estimated from the study population's genotypes, which corresponds to a normalized Euclidean distance kernel. In this paper, we show that reliance on the Euclidean distance kernel contributes to several unresolved modeling inconsistencies in heritability estimation for GWAS. These inconsistencies can cause biased heritability estimates in the presence of linkage disequilibrium (LD), depending on the distribution of causal variants. We show that these biases can be resolved (at least at the modeling level) if one adopts a Mahalanobis distance-based GRM for LMM analysis. Additionally, we propose a new definition of partitioned heritability -- the heritability attributable to a subset of genes or single nucleotide polymorphisms (SNPs) -- using the Mahalanobis GRM, and show that it inherits many of the nice consistency properties identified in our original analysis. Partitioned heritability is a relatively new area for GWAS analysis, where inconsistency issues related to LD have previously been known to be especially pernicious.

stat.AP↗

arXiv2018

Approximate nonparametric maximum likelihood inference for mixture models via convex optimization

Nonparametric maximum likelihood (NPML) for mixture models is a technique for estimating mixing distributions that has a long and rich history in statistics going back to the 1950s, and is closely related to empirical Bayes methods. Historically, NPML-based methods have been considered to be relatively impractical because of computational and theoretical obstacles. However, recent work focusing on approximate NPML methods suggests that these methods may have great promise for a variety of modern applications. Building on this recent work, a class of flexible, scalable, and easy to implement approximate NPML methods is studied for problems with multivariate mixing distributions. Concrete guidance on implementing these methods is provided, with theoretical and empirical support; topics covered include identifying the support set of the mixing distribution, and comparing algorithms (across a variety of metrics) for solving the simple convex optimization problem at the core of the approximate NPML problem. Additionally, three diverse real data applications are studied to illustrate the methods' performance: (i) A baseball data analysis (a classical example for empirical Bayes methods), (ii) high-dimensional microarray classification, and (iii) online prediction of blood-glucose density for diabetes patients. Among other things, the empirical results demonstrate the relative effectiveness of using multivariate (as opposed to univariate) mixing distributions for NPML-based approaches.

stat.ME↗

arXiv2016

Scalable Approximations for Generalized Linear Problems

In stochastic optimization, the population risk is generally approximated by the empirical risk. However, in the large-scale setting, minimization of the empirical risk may be computationally restrictive. In this paper, we design an efficient algorithm to approximate the population risk minimizer in generalized linear problems such as binary classification with surrogate losses and generalized linear regression models. We focus on large-scale problems, where the iterative minimization of the empirical risk is computationally intractable, i.e., the number of observations $n$ is much larger than the dimension of the parameter $p$, i.e. $n \gg p \gg 1$. We show that under random sub-Gaussian design, the true minimizer of the population risk is approximately proportional to the corresponding ordinary least squares (OLS) estimator. Using this relation, we design an algorithm that achieves the same accuracy as the empirical risk minimizer through iterations that attain up to a cubic convergence rate, and that are cheaper than any batch optimization algorithm by at least a factor of $\mathcal{O}(p)$. We provide theoretical guarantees for our algorithm, and analyze the convergence behavior in terms of data dimensions. Finally, we demonstrate the performance of our algorithm on well-known classification and regression problems, through extensive numerical studies on large-scale datasets, and show that it achieves the highest performance compared to several other widely used and specialized optimization algorithms.

stat.ML↗

arXiv2016

Kernel ridge vs. principal component regression: minimax bounds and adaptability of regularization operators

Regularization is an essential element of virtually all kernel methods for nonparametric regression problems. A critical factor in the effectiveness of a given kernel method is the type of regularization that is employed. This article compares and contrasts members from a general class of regularization techniques, which notably includes ridge regression and principal component regression. We derive an explicit finite-sample risk bound for regularization-based estimators that simultaneously accounts for (i) the structure of the ambient function space, (ii) the regularity of the true regression function, and (iii) the adaptability (or qualification) of the regularization. A simple consequence of this upper bound is that the risk of the regularization-based estimators matches the minimax rate in a variety of settings. The general bound also illustrates how some regularization techniques are more adaptable than others to favorable regularity properties that the true regression function may possess. This, in particular, demonstrates a striking difference between kernel ridge regression and kernel principal component regression. Our theoretical results are supported by numerical experiments.

math.ST↗

arXiv2016

Ridge regression and asymptotic minimax estimation over spheres of growing dimension

We study asymptotic minimax problems for estimating a $d$-dimensional regression parameter over spheres of growing dimension ($d\to \infty$). Assuming that the data follows a linear model with Gaussian predictors and errors, we show that ridge regression is asymptotically minimax and derive new closed form expressions for its asymptotic risk under squared-error loss. The asymptotic risk of ridge regression is closely related to the Stieltjes transform of the Marčenko-Pastur distribution and the spectral distribution of the predictors from the linear model. Adaptive ridge estimators are also proposed (which adapt to the unknown radius of the sphere) and connections with equivariant estimation are highlighted. Our results are mostly relevant for asymptotic settings where the number of observations, $n$, is proportional to the number of predictors, that is, $d/n\toρ\in(0,\infty)$.

math.ST↗

arXiv2015

Flexible results for quadratic forms with applications to variance components estimation

We derive convenient uniform concentration bounds and finite sample multivariate normal approximation results for quadratic forms, then describe some applications involving variance components estimation in linear random-effects models. Random-effects models and variance components estimation are classical topics in statistics, with a corresponding well-established asymptotic theory. However, our finite sample results for quadratic forms provide additional flexibility for easily analyzing random-effects models in non-standard settings, which are becoming more important in modern applications (e.g. genomics). For instance, in addition to deriving novel non-asymptotic bounds for variance components estimators in classical linear random-effects models, we provide a concentration bound for variance components estimators in linear models with correlated random-effects. Our general concentration bound is a uniform version of the Hanson-Wright inequality. The main normal approximation result in the paper is derived using Reinert and Röllin's (2009) embedding technique and multivariate Stein's method with exchangeable pairs.

math.ST↗

arXiv2014

Nonparametric empirical Bayes and maximum likelihood estimation for high-dimensional data analysis

Nonparametric empirical Bayes methods provide a flexible and attractive approach to high-dimensional data analysis. One particularly elegant empirical Bayes methodology, involving the Kiefer-Wolfowitz nonparametric maximum likelihood estimator (NPMLE) for mixture models, has been known for decades. However, implementation and theoretical analysis of the Kiefer-Wolfowitz NPMLE are notoriously difficult. A fast algorithm was recently proposed that makes NPMLE-based procedures feasible for use in large-scale problems, but the algorithm calculates only an approximation to the NPMLE. In this paper we make two contributions. First, we provide upper bounds on the convergence rate of the approximate NPMLE's statistical error, which have the same order as the best known bounds for the true NPMLE. This suggests that the approximate NPMLE is just as effective as the true NPMLE for statistical applications. Second, we illustrate the promise of NPMLE procedures in a high-dimensional binary classification problem. We propose a new procedure and show that it vastly outperforms existing methods in experiments with simulated data. In real data analyses involving cancer survival and gene expression data, we show that it is very competitive with several recently proposed methods for regularized linear discriminant analysis, another popular approach to high-dimensional classification.

stat.ME↗

arXiv2012

Residual variance and the signal-to-noise ratio in high-dimensional linear models

Residual variance and the signal-to-noise ratio are important quantities in many statistical models and model fitting procedures. They play an important role in regression diagnostics, in determining the performance limits in estimation and prediction problems, and in shrinkage parameter selection in many popular regularized regression methods for high-dimensional data analysis. We propose new estimators for the residual variance, the l2-signal strength, and the signal-to-noise ratio that are consistent and asymptotically normal in high-dimensional linear models with Gaussian predictors and errors, where the number of predictors d is proportional to the number of observations n. Existing results on residual variance estimation in high-dimensional linear models depend on sparsity in the underlying signal. Our results require no sparsity assumptions and imply that the residual variance may be consistently estimated even when d > n and the underlying signal itself is non-estimable. Basic numerical work suggests that some of the distributional assumptions made for our theoretical results may be relaxed.

stat.ME↗