Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as well as special models (e.g. Toeplitz, sparse, trees) that are of interest in statistics. We study the maximum likelihood degree and its dual analogue, and we introduce a new software package LinearCovarianceModels.jl for solving the score equations. All local maxima can thus be computed reliably. In addition we identify several scenarios for which the estimator is a rational function.
The estimation of covariance matrices of gene expressions has many applications in cancer systems biology. Many gene expression studies, however, are hampered by low sample size and it has therefore become popular to increase sample size by collecting gene expression data across studies. Motivated by the traditional meta-analysis using random effects models, we present a hierarchical random covariance model and use it for the meta-analysis of gene correlation networks across 11 large-scale gene expression studies of diffuse large B-cell lymphoma (DLBCL). We suggest to use a maximum likelihood …
This paper introduces a general framework for estimating variance components in the linear mixed models via general unbiased estimating equations, which include some well-used estimators such as the restricted maximum likelihood estimator. We derive the asymptotic covariance matrices and second-order biases under general estimating equations without assuming the normality of the underlying distributions and identify a class of second-order unbiased estimators of variance components. It is also shown that the asymptotic covariance matrices and second-order biases do not depend on whether the re…
We propose a new estimator for the spot covariance matrix of a multi-dimensional continuous semi-martingale log asset price process which is subject to noise and non-synchronous observations. The estimator is constructed based on a local average of block-wise parametric spectral covariance estimates. The latter originate from a local method of moments (LMM) which recently has been introduced. We prove consistency and a point-wise stable central limit theorem for the proposed spot covariance estimator in a very general setup with stochastic volatility, leverage effects and general noise distrib…
The inverse covariance matrix provides considerable insight for understanding statistical models in the multivariate setting. In particular, when the distribution over variables is assumed to be multivariate normal, the sparsity pattern in the inverse covariance matrix, commonly referred to as the precision matrix, corresponds to the adjacency matrix representation of the Gauss-Markov graph, which encodes conditional independence statements between variables. Minimax results under the spectral norm have previously been established for covariance matrices, both sparse and banded, and for sparse…
The problem of estimating the spectral density matrix $f(w)$ of a multivariate time series is revisited with special focus on the frequencies $w=0$ and $w=π$. Recognizing that the entries of the spectral density matrix at these two boundary points are real-valued, we propose a new estimator constructed from a local polynomial regression of the real portion of the multivariate periodogram. The case $w=0$ is of particular importance, since $f(0)$ is associated with the large-sample covariance matrix of the sample mean; hence, estimating $f(0)$ is crucial in order to conduct any sort of statistic…
Multiple hypothesis testing is a fundamental problem in high dimensional inference, with wide applications in many scientific fields. In genome-wide association studies, tens of thousands of tests are performed simultaneously to find if any SNPs are associated with some traits and those tests are correlated. When test statistics are correlated, false discovery control becomes very challenging under arbitrary dependence. In the current paper, we propose a novel method based on principal factor approximation, which successfully subtracts the common dependence and weakens significantly the correl…
While most of the convergence results in the literature on high dimensional covariance matrix are concerned about the accuracy of estimating the covariance matrix (and precision matrix), relatively less is known about the effect of estimating large covariances on statistical inferences. We study two important models: factor analysis and panel data model with interactive effects, and focus on the statistical inference and estimation efficiency of structural parameters based on large covariance estimators. For efficient estimation, both models call for a weighted principle components (WPC), whic…
We elucidate the problem of estimating large-dimensional covariance matrices in the presence of correlations between samples. To this end, we generalize the Marcenko-Pastur equation and the Ledoit-Peche shrinkage estimator using methods of random matrix theory and free probability. We develop an efficient algorithm that implements the corresponding analytic formulas, based on the Ledoit-Wolf kernel estimation technique. We also provide an associated open-source Python library, called "shrinkage", with a user-friendly API to assist in practical tasks of estimation of large covariance matrices. …
Estimating the unconstrained mean and covariance matrix is a popular topic in statistics. However, estimation of the parameters of $N_p(μ,Σ)$ under joint constraints such as $Σμ= μ$ has not received much attention. It can be viewed as a multivariate counterpart of the classical estimation problem in the $N(θ,θ^2)$ distribution. In addition to the usual inference challenges under such non-linear constraints among the parameters (curved exponential family), one has to deal with the basic requirements of symmetry and positive definiteness when estimating a covariance matrix. We derive the non-lin…
This paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and Bregman divergence losses. A major focus is on the derivation of a rate sharp minimax lower bound. The problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems. Standard techniques fail to yield good results, and new tools are thus needed. We first develop a lower bound technique that is particularly well suited for treating "two-directional" problems suc…
We prove new lower bounds for statistical estimation tasks under the constraint of $(\varepsilon, δ)$-differential privacy. First, we provide tight lower bounds for private covariance estimation of Gaussian distributions. We show that estimating the covariance matrix in Frobenius norm requires $Ω(d^2)$ samples, and in spectral norm requires $Ω(d^{3/2})$ samples, both matching upper bounds up to logarithmic factors. The latter bound verifies the existence of a conjectured statistical gap between the private and the non-private sample complexities for spectral estimation of Gaussian covariances.…
Estimating spot covariance is an important issue to study, especially with the increasing availability of high-frequency financial data. We study the estimation of spot covariance using a kernel method for high-frequency data. In particular, we consider first the kernel weighted version of realized covariance estimator for the price process governed by a continuous multivariate semimartingale. Next, we extend it to the threshold kernel estimator of the spot covariances when the underlying price process is a discontinuous multivariate semimartingale with finite activity jumps. We derive the asy…
The accurate computation of the covariance matrix of fitted model parameters is a somewhat neglected task in Statistics. Algorithms are given for computing accurate covariance matrices derived from computing the Hessian matrix by numerical differentiation, and also for the covariance matrix of the posterior distribution of model parameters. Evaluations on two datasets where the Hessian could be computed analytically show that the numerical differentiation algorithm is very accurate.
Lag windows are commonly used in time series, econometrics, steady-state simulation, and Markov chain Monte Carlo to estimate time-average covariance matrices. In the presence of positive correlation of the underlying process, estimators of this matrix almost always exhibit significant negative bias, leading to undesirable finite-sample properties. We propose a new family of lag windows specifically designed to improve finite-sample performance by offsetting this negative bias. Any existing lag window can be adapted into a lugsail equivalent with no additional assumptions. We use these lag win…
Quantitative portfolio allocation requires the accurate and tractable estimation of covariances between a large number of assets, whose histories can greatly vary in length. Such data are said to follow a monotone missingness pattern, under which the likelihood has a convenient factorization. Upon further assuming that asset returns are multivariate normally distributed, with histories at least as long as the total asset count, maximum likelihood (ML) estimates are easily obtained by performing repeated ordinary least squares (OLS) regressions, one for each asset. Things get more interesting w…
Bootstrap methods for estimating the long-run covariance of stationary functional time series are considered. We introduce a versatile bootstrap method that relies on functional principal component analysis, where principal component scores can be bootstrapped by maximum entropy. Two other bootstrap methods resample error functions, after the dependence structure being modeled linearly by a sieve method or nonlinearly by a functional kernel regression. Through a series of Monte-Carlo simulation, we evaluate and compare the finite-sample performances of these three bootstrap methods for estimat…
This paper provides a framework for estimating the mean and variance of a high-dimensional normal density. The main setting considered is a fixed number of vector following a high-dimensional normal distribution with unknown mean and diagonal covariance matrix. The diagonal covariance matrix can be known or unknown. If the covariance matrix is unknown, the sample size can be as small as $2$. The proposed estimator is based on the idea that the unobserved pairs of mean and variance for each dimension are drawn from an unknown bivariate distribution, which we model as a mixture of normal-inverse…
We develop a multi-level restricted Gaussian maximum likelihood method for estimating the covariance function parameters and computing the best unbiased predictor. Our approach produces a new set of multi-level contrasts where the deterministic parameters of the model are filtered out thus enabling the estimation of the covariance parameters to be decoupled from the deterministic component. Moreover, the multi-level covariance matrix of the contrasts exhibit fast decay that is dependent on the smoothness of the covariance function. Due to the fast decay of the multi-level covariance matrix coe…
We consider reduced-rank modeling of the white noise covariance matrix in a large dimensional vector autoregressive (VAR) model. We first propose the reduced-rank covariance estimator under the setting where independent observations are available. We derive the reduced-rank estimator based on a latent variable model for the vector observation and give the analytical form of its maximum likelihood estimate. Simulation results show that the reduced-rank covariance estimator outperforms two competing covariance estimators for estimating large dimensional covariance matrices from independent obser…
In spatial statistics, it is often assumed that the spatial field of interest is stationary and its covariance has a simple parametric form, but these assumptions are not appropriate in many applications. Given replicate observations of a Gaussian spatial field, we propose nonstationary and nonparametric Bayesian inference on the spatial dependence. Instead of estimating the quadratic (in the number of spatial locations) entries of the covariance matrix, the idea is to infer a near-linear number of nonzero entries in a sparse Cholesky factor of the precision matrix. Our prior assumptions are m…
