This paper considers testing the covariance matrices structure based on Wald's score test in large dimensional setting. The hypothesis $H_0: Σ=Σ_0 $ for a given matrix $Σ_0$, which covers the identity hypothesis test and sphericity hypothesis test as the special cases, is reviewed by the generalized CLT (Central Limit Theorem) for the linear spectral statistics of large dimensional sample covariance matrices from Jiang(2015) . The proposed tests can be applicable for large dimensional non-Gaussian variables in a wider range. Furthermore, the simulation study is provided to compare the proposed tests with other large dimensional covariance matrix tests for evaluation of their performances. As seen from the simulation results, our proposed tests are feasible for large dimensional data without restriction of population distribution and provide the accurate and steady empirical sizes, which are almost around the nominal size.
We propose a new space of phylogenetic trees which we call wald space. The motivation is to develop a space suitable for statistical analysis of phylogenies, but with a geometry based on more biologically principled assumptions than existing spaces: in wald space, trees are close if they induce similar distributions on genetic sequence data. As a point set, wald space contains the previously developed Billera-Holmes-Vogtmann (BHV) tree space; it also contains disconnected forests, like the edge-product (EP) space but without certain singularities of the EP space. We investigate two related geo…
This paper considers the optimal modification of the likelihood ratio test (LRT) for the equality of two high-dimensional covariance matrices. The classical LRT is not well defined when the dimensions are larger than or equal to one of the sample sizes. In this paper, an optimally modified test that works well in cases where the dimensions may be larger than the sample sizes is proposed. In addition, the test is established under the weakest conditions on the moments and the dimensions of the samples. We also present weakly consistent estimators of the fourth moments, which are necessary for t…
The main theme of this paper is a modification of the likelihood ratio test (LRT) for testing high dimensional covariance matrix. Recently, the correct asymptotic distribution of the LRT for a large-dimensional case (the case $p/n$ approaches to a constant $γ\in (0,1]$) is specified by researchers. The correct procedure is named as corrected LRT. Despite of its correction, the corrected LRT is a function of sample eigenvalues that are suffered from redundant variability from high dimensionality and, subsequently, still does not have full power in differentiating hypotheses on the covariance ma…
Standard statistical methods that do not take proper account of the complexity of survey design can lead to erroneous inferences when applied to survey data due to unequal selection probabilities, clustering, and other design features. In particular, the actual type I error rates of tests of hypotheses using standard methods can be much bigger than the nominal significance level. Methods that take account of survey design features in testing hypotheses have been proposed, including Wald tests and quasi-score tests that involve the estimated covariance matrices of parameter estimates. In this p…
This paper provides an entire inference procedure for the autoregressive model under (conditional) heteroscedasticity of unknown form with a finite variance. We first establish the asymptotic normality of the weighted least absolute deviations estimator (LADE) for the model. Second, we develop the random weighting (RW) method to estimate its asymptotic covariance matrix, leading to the implementation of the Wald test. Third, we construct a portmanteau test for model checking, and use the RW method to obtain its critical values. As a special weighted LADE, the feasible adaptive LADE (ALADE) is …
Large-scale multiple testing with highly correlated test statistics arises frequently in many scientific research. Incorporating correlation information in estimating false discovery proportion has attracted increasing attention in recent years. When the covariance matrix of test statistics is known, Fan, Han & Gu (2012) provided a consistent estimate of False Discovery Proportion (FDP) under arbitrary dependence structure. However, the covariance matrix is often unknown in many applications and such dependence information has to be estimated before estimating FDP (Efron, 2010). The estimation…
Missing data estimation is an important challenge with high-dimensional data arranged in the form of a matrix. Typically this data matrix is transposable, meaning that either the rows, columns or both can be treated as features. To model transposable data, we present a modification of the matrix-variate normal, the mean-restricted matrix-variate normal, in which the rows and columns each have a separate mean vector and covariance matrix. By placing additive penalties on the inverse covariance matrices of the rows and columns, these so-called transposable regularized covariance models allow for…
The question of testing for equality in distribution between two linear models, each consisting of sums of distinct discrete independent random variables with unequal numbers of observations, has emerged from the biological research. In this case, the computation of classical $χ^2$ statistics, which would not include all observations, results in loss of power, especially when sample sizes are small. Here, as an alternative that uses all data, the nonparametric maximum likelihood estimator for the distribution of sum of discrete and independent random variables, which we call the convolution st…
In this work, we study the positive definiteness (PDness) problem in covariance matrix estimation. For high dimensional data, many regularized estimators are proposed under structural assumptions on the true covariance matrix including sparsity. They are shown to be asymptotically consistent and rate-optimal in estimating the true covariance matrix and its structure. However, many of them do not take into account the PDness of the estimator and produce a non-PD estimate. To achieve the PDness, researchers consider additional regularizations (or constraints) on eigenvalues, which make both the …
The L1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to recent state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modific…
Fitting high-dimensional data involves a delicate tradeoff between faithful representation and the use of sparse models. Too often, sparsity assumptions on the fitted model are too restrictive to provide a faithful representation of the observed data. In this paper, we present a novel framework incorporating sparsity in different domains.We decompose the observed covariance matrix into a sparse Gaussian Markov model (with a sparse precision matrix) and a sparse independence model (with a sparse covariance matrix). Our framework incorporates sparse covariance and sparse precision estimation as …
Inference about a scalar parameter of interest typically relies on the asymptotic normality of common likelihood pivots, such as the signed likelihood root, the score and Wald statistics. Nevertheless, the resulting inferential procedures are known to perform poorly when the dimension of the nuisance parameter is large relative to the sample size and when the information about the parameters is limited. In many such cases, the use of asymptotic normality of analytical modifications of the signed likelihood root is known to recover inferential performance. It is proved here that parametric boot…
In this paper, we present a novel framework incorporating a combination of sparse models in different domains. We posit the observed data as generated from a linear combination of a sparse Gaussian Markov model (with a sparse precision matrix) and a sparse Gaussian independence model (with a sparse covariance matrix). We provide efficient methods for decomposition of the data into two domains, \viz Markov and independence domains. We characterize a set of sufficient conditions for identifiability and model consistency. Our decomposition method is based on a simple modification of the popular $…
All models may be wrong -- but that is not necessarily a problem for inference. Consider the standard $t$-test for the significance of a variable $X$ for predicting response $Y$ whilst controlling for $p$ other covariates $Z$ in a random design linear model. This yields correct asymptotic type~I error control for the null hypothesis that $X$ is conditionally independent of $Y$ given $Z$ under an \emph{arbitrary} regression model of $Y$ on $(X, Z)$, provided that a linear regression model for $X$ on $Z$ holds. An analogous robustness to misspecification, which we term the "double-estimation-fri…
Clinical trials are common in medical research where multiple non-Gaussian responses and time-dependent observations are frequent. The analysis of data from these studies requires statistical modeling techniques that take these characteristics into account. We propose a general strategy based on the Wald statistics to perform hypothesis tests like ANOVAs, MANOVAs and multiple comparison tests on regression and dispersion parameters of multivariate covariance generalized linear models (McGLMs). McGLMs provide a general statistical modeling framework for normal and non-normal multivariate data a…
Highly robust factored principal component analysis for matrix-valued outlier accommodation and explainable detection via matrix minimum covariance determinant
In this paper we consider the use of the space vs. time Kronecker product decomposition in the estimation of covariance matrices for spatio-temporal data. This decomposition imposes lower dimensional structure on the estimated covariance matrix, thus reducing the number of samples required for estimation. To allow a smooth tradeoff between the reduction in the number of parameters (to reduce estimation variance) and the accuracy of the covariance approximation (affecting estimation bias), we introduce a diagonally loaded modification of the sum of kronecker products representation [1]. We deri…
For a high-dimensional parameter of interest, tests based on quadratic statistics are known to have low power against subsets of the parameter space (henceforth, parameter subspaces). In addition, they typically involve an inverse covariance matrix which is difficult to estimate in high-dimensional settings. I simultaneously address these two issues by proposing a novel test statistic that is large in a conic parameter subspace of interest. This test statistic generalizes the Wald statistic and nests many well-known test statistics. For a given parameter subspace, the statistic is free of tuni…
