Asymptotic Covariance Matrix Research Articles

Designing three-level cluster randomized trials to assess treatment effect heterogeneity.

Cluster randomized trials often exhibit a three-level structure with participants nested in subclusters such as health care providers, and subclusters nested in clusters such as clinics. While the average treatment effect has been the primary focus in planning three-level randomized trials, interest is growing in understanding whether the treatment effect varies among prespecified patient subpopulations, such as those defined by demographics or baseline clinical characteristics. In this article, we derive novel analytical design formulas based on the asymptotic covariance matrix for powering confirmatory analyses of treatment effect heterogeneity in three-level trials, that are broadly applicable to the evaluation of cluster-level, subcluster-level, and participant-level effect modifiers and to designs where randomization can be carried out at any level. We characterize a nested exchangeable correlation structure for both the effect modifier and the outcome conditional on the effect modifier, and generate new insights from a study design perspective for conducting analyses of treatment effect heterogeneity based on a linear mixed analysis of covariance model. A simulation study is conducted to validate our new methods and two real-world trial examples are used for illustrations.

Estimation of ergodic square-root diffusion under high-frequency sampling

Gaussian quasi-likelihood estimation of the parameter in the square-root diffusion process is studied under high-frequency sampling. Different from previous studies under low-frequency sampling, high-frequency of data leads to a very simple form of the asymptotic covariance matrix. Through easy-to-compute preliminary contrast functions, a practical two-stage manner without numerical optimization is formulated to conduct not only an asymptotically efficient estimation of the drift parameters but also a high-precision estimator of the diffusion parameter. Simulation experiments are given to illustrate the results.

Simple: Statistical Inference on Membership Profiles in Large Networks

AbstractNetwork data are prevalent in many contemporary big data applications in which a common interest is to unveil important latent links between different pairs of nodes. Yet a simple fundamental question of how to precisely quantify the statistical uncertainty associated with the identification of latent links still remains largely unexplored. In this paper, we propose the method of statistical inference on membership profiles in large networks (SIMPLE) in the setting of degree-corrected mixed membership model, where the null hypothesis assumes that the pair of nodes share the same profile of community memberships. In the simpler case of no degree heterogeneity, the model reduces to the mixed membership model for which an alternative more robust test is also proposed. Both tests are of the Hotelling-type statistics based on the rows of empirical eigenvectors or their ratios, whose asymptotic covariance matrices are very challenging to derive and estimate. Nevertheless, their analytical expressions are unveiled and the unknown covariance matrices are consistently estimated. Under some mild regularity conditions, we establish the exact limiting distributions of the two forms of SIMPLE test statistics under the null hypothesis and contiguous alternative hypothesis. They are the chi-square distributions and the noncentral chi-square distributions, respectively, with degrees of freedom depending on whether the degrees are corrected or not. We also address the important issue of estimating the unknown number of communities and establish the asymptotic properties of the associated test statistics. The advantages and practical utility of our new procedures in terms of both size and power are demonstrated through several simulation examples and real network applications.

Asymptotic covariance estimation by Gaussian random perturbation

In most cases, the asymptotic covariance matrix of an M-estimator is in a sandwich form. This sandwich form involves calculations of the first and second order derivatives of the loss function, which is intractable if the loss function is complex. To alleviate this problem, we propose in this article a novel method called Gaussian random perturbation. This method can be used to estimate the asymptotic covariance matrix of a general M-estimator without derivative calculations. The idea can be summarized as follows. We first generate a small random perturbation around the M-estimator. Then, we re-evaluate the loss function at the randomly perturbed M-estimator and obtain the estimators of the first and second order derivatives of the loss function via Taylor series expansion. This leads to a novel estimator for the asymptotic covariance matrix. We then rigorously show that the resulting covariance estimator is statistically consistent with two elegant characteristics. First, it involves no computation of derivatives. This makes it easier to estimate the covariance matrix of an M-estimator with a complex loss function. Second, it is convenient for parallel computing and thus attractive for massive data analysis. The consistency of the proposed asymptotic covariance estimator is demonstrated under appropriate regularity conditions. The practical usefulness of the method is further demonstrated with both simulation studies and real data analysis.

Novel Approach of Multistate Markov Chains to Evaluate Progression in the Expanded Model of Non-alcoholic Fatty Liver Disease

A global increase in the prevalence of obesity and type 2 diabetes is strongly connected to an increased prevalence of non-alcoholic fatty liver disease (NAFLD) worldwide. In this article, the progression of the NAFLD process is modeled by continuous time Markov chains (CTMCs) with nine states. Maximum likelihood is used to estimate the transition intensities among the states. Once the transition intensities are obtained, the mean sojourn time and its variance are estimated, and the state probability distribution and its asymptotic covariance matrix are also estimated. A hypothetical example based on a longitudinal study assessing patients with NAFLD in various stages is discussed. The mean time to absorption is estimated, and the other abovementioned statistical indices are examined. In this article, the maximum likelihood estimation (MLE) function is utilized in a new approach to compensate for the missing values in the follow-up period of patients evaluated in longitudinal studies. A MATLAB code link is provided, at the end of the article, for the estimation of the transition rate matrix and transition probability matrix.

Globally Centered Autocovariances in MCMC

Autocovariances are a fundamental quantity of interest in Markov chain Monte Carlo (MCMC) simulations with autocorrelation function (ACF) plots being an integral visualization tool for performance assessment. Unfortunately, for slow-mixing Markov chains, the empirical autocovariance can highly underestimate the truth. For multiple-chain MCMC sampling, we propose a globally centered estimator of the autocovariance function (G-ACvF) that exhibits significant theoretical and empirical improvements. We show that the bias of the G-ACvF estimator is smaller than the bias of the current state-of-the-art. The impact of this improved estimator is evident in three critical output analysis applications: (a) ACF plots, (b) estimates of the Monte Carlo asymptotic covariance matrix, and (c) estimates of the effective sample size. Under weak conditions, we establish strong consistency of our improved asymptotic covariance estimator, and obtain its large-sample bias and variance. The performance of the new estimators is demonstrated through various examples. Supplementary materials for this article are available online.

Feasible Panel GARCH Models: Variance-Targeting Estimation and Empirical Application

For the panel generalized autoregressive conditional heteroskedasticity (GARCH) model, the conditions for the stationarity and positive definiteness of conditional covariance processes are examined. A new feasible specification is constructed for the class of panel GARCH models, and a three-step estimation technique is developed based on a variance-targeting (VT) approach. The consistency and asymptotic normality of the VT estimator are shown when the time dimension tends to infinity and the cross-sectional dimension is fixed. The stationarity and asymptotic properties are discussed for both time and cross-sectional dimensions tend to infinity. The results of Monte Carlo experiments indicate that the finite sample property of the VT estimator is satisfactory, implying that increasing the cross-sectional dimension does not affect the speed of convergence, but shrinks the asymptotic covariance matrix. The empirical results of the analysis of the inflation rates of G7 countries and growth rates for the value of trade in four economic regions indicate that the feasible specification provides a competitive alternative to the class of panel GARCH models. The empirical results indicate that the global financial crisis affects the growth rates of trades, while the influence of the COVID-19 pandemic shows that its effect on inflation rates is insignificant.

Jackknife for nonlinear estimating equations

In mixture with varying concentrations model (MVC) one deals with a nonhomogeneous sample which consists of subjects belonging to a fixed number of different populations (mixture components). The population which a subject belongs to is unknown, but the probabilities to belong to a given component are known and vary from observation to observation. The distribution of subjects’ observed features depends on the component which it belongs to. Generalized estimating equations (GEE) for Euclidean parameters in MVC models are considered. Under suitable assumptions the obtained estimators are asymptotically normal. A jackknife (JK) technique for the estimation of their asymptotic covariance matrices is described. Consistency of JK-estimators is demonstrated. An application to a model of mixture of nonlinear regressions and a real life example are presented.

Scale-Wise Variance Minimization for Optimal Virtual Signals: An Approach for Redundant Gyroscopes

The increased use of low-cost gyroscopes within inertial sensors for navigation purposes, among others, has brought to the development of a considerable amount of research in improving their measurement precision. Aside from developing methods that allow to model and account for the deterministic and stochastic components that contribute to the measurement errors of these devices, an approach that has been put forward in recent years is to make use of arrays of such sensors in order to combine their measurements thereby reducing the impact of individual sensor noise. Nevertheless combining these measurements is not straightforward given the complex stochastic nature of these errors and, although some solutions have been suggested, these are limited to certain specific settings which do not allow to achieve solutions in more general and common circumstances. Hence, in this work we put forward a non-parametric method that makes use of the wavelet cross-covariance at different scales to combine the measurements coming from an array of gyroscopes in order to deliver an optimal measurement signal without needing any assumption on the processes underlying the individual error signals. As a result of this work we also study an appropriate non-parametric approach for the estimation of the asymptotic covariance matrix of the wavelet cross-covariance estimator which has important applications beyond the scope of this work. The theoretical properties of the proposed approach are studied and are supported by simulations and real applications, indicating that this method represents an appropriate and general tool for the construction of optimal virtual signals that are particularly relevant for arrays of gyroscopes. Moreover, the results of this work can support the creation of optimal signals for other types of inertial sensors other than gyroscopes as well as for redundant measurements in other domains other than navigation.

Effect of measurement error size in linear heteroscedastic measurement error models

Measurement errors affect the properties of obtained optimal estimators. Further, they affect the accuracy of estimation, although the size of the measurement error required to invalidate an estimation method remains unknown. In this study, we described the effects of measurement errors on estimation and the derived asymptotic covariance matrix by using the properties of unbiased estimating equations. We evaluated different estimation methodologies using various combinations of sample sizes and average reliability ratios. Simulation studies have compared estimators in terms of simulated bias, mean squared error, and coverage probability for the slope parameter. By comparing estimation approaches in terms of bias, mean squared error, and coverage probability, we found that the average reliability ratio of 0.65 and a sample size of 20 yielded estimator stability, with only slight differences in these three attributes. If the sample size was increased, the cutoff point for the average reliability ratio could be relaxed. This process has been in this study illustrated using real datasets.

Moment estimators for parameters of Lévy‐driven Ornstein–Uhlenbeck processes

We consider the problem of parameter estimation for Ornstein–Uhlenbeck (OU) processes driven by general Lévy processes. We derive our estimators based on the method of moments and establish a joint central limit theorem for these estimators with explicit formulae for their asymptotic covariance matrix. Numerical experiments are also provided to show that not only our estimators are easy to implement but they are also highly efficient. Our work offers a simple and efficient method to estimate the parameters in Lévy‐driven OU processes.

The Extended Birnbaum\u2013Saunders Distribution Based on the Scale Shape Mixture of Skew Normal Distributions

In this article, a large class of univriate Birnbaum–Saunders distributions based on the scale shape mixture of skew normal distributions is introduced which generates suitable subclasses for modeling asymmetric data in a variety of settings. The moments and maximum likelihood estimation procedures are disscused via an ECM-algorithm. The observed information matrix to approximate the asymptotic covariance matrix of the parameter estimates is then derived in some subclasses. A simulation study is also performed to evaluate the finite sample properties of ML estimators and finally, a real data set is analyzed for illustrative purposes.

Nonlinear multiplicative distortion regression models with second-order estimation

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. We proposed the second order least squares estimator by including in the criterion function the distance of both the response variable to its conditional mean and the squared response variable to its second conditional moment given the predictor variables. The asymptotic results of the second order nonlinear least squares estimators for the regression parameter are derived. As a by-product, we construct confidence intervals by estimating the asymptotic covariance matrices. We also consider to use three existing calibration procedures to calibrate the unknown response and covariates, namely, the conditional absolute mean calibration, the conditional variance calibration and the logarithmic calibration. The simulation studies shows that these second order nonlinear least squares estimators with three calibration procedures are all superior than the ordinary nonlinear least squares estimators.

Estimating the variance of a combined forecast: Bootstrap-based approach

This paper considers bootstrap inference in model averaging for predictive regressions. We first show that the standard pairwise bootstrap is not valid in the context of model averaging. This common bootstrap approach induces a bias-related term in the bootstrap variance of averaging estimators. We then propose and justify a fixed-design residual-based bootstrap resampling approach for model averaging. In a local asymptotic framework, we show the validity of the bootstrap in estimating the variance of a combined forecast and the asymptotic covariance matrix of a combined parameter vector with fixed weights. Our proposed method preserves non-parametrically the cross-sectional dependence between different models and the time series dependence in the errors simultaneously. The finite sample performance of these methods is assessed via Monte Carlo simulations. We illustrate our approach using an empirical study of the Taylor rule equation with 24 alternative specifications.

Covariances in Pólya urn schemes

AbstractBy now there is a solid theory for Polya urns. Finding the covariances is somewhat laborious. While these papers are “structural,” our purpose here is “computational.” We propose a practicable method for building the asymptotic covariance matrix in tenable balanced urn schemes, whereupon the asymptotic covariance matrix is obtained by solving a linear system of equations. We demonstrate the use of the method in growing tenable balanced irreducible schemes with a small index and in critical urns. In the critical case, the solution to the linear system of equations is explicit in terms of an eigenvector of the scheme.

Two-phase sample selection strategies for design and analysis in post-genome-wide association fine-mapping studies.

Post‐GWAS analysis, in many cases, focuses on fine‐mapping targeted genetic regions discovered at GWAS‐stage; that is, the aim is to pinpoint potential causal variants and susceptibility genes for complex traits and disease outcomes using next‐generation sequencing (NGS) technologies. Large‐scale GWAS cohorts are necessary to identify target regions given the typically modest genetic effect sizes. In this context, two‐phase sampling design and analysis is a cost‐reduction technique that utilizes data collected during phase 1 GWAS to select an informative subsample for phase 2 sequencing. The main goal is to make inference for genetic variants measured via NGS by efficiently combining data from phases 1 and 2. We propose two approaches for selecting a phase 2 design under a budget constraint. The first method identifies sampling fractions that select a phase 2 design yielding an asymptotic variance covariance matrix with certain optimal characteristics, for example, smallest trace, via Lagrange multipliers (LM). The second relies on a genetic algorithm (GA) with a defined fitness function to identify exactly a phase 2 subsample. We perform comprehensive simulation studies to evaluate the empirical properties of the proposed designs for a genetic association study of a quantitative trait. We compare our methods against two ranked designs: residual‐dependent sampling and a recently identified optimal design. Our findings demonstrate that the proposed designs, GA in particular, can render competitive power in combined phase 1 and 2 analysis compared with alternative designs while preserving type 1 error control. These results are especially evident under the more practical scenario where design values need to be defined a priori and are subject to misspecification. We illustrate the proposed methods in a study of triglyceride levels in the North Finland Birth Cohort of 1966. R code to reproduce our results is available at github.com/egosv/TwoPhase_postGWAS.

General unbiased estimating equations for variance components in linear mixed models

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 regression coefficients are estimated by the generalized or ordinary least squares methods. We carry out numerical studies to check the performance of the proposed methods based on typical linear mixed models.

Inference on maintenance service policy under step-stress partially accelerated life tests using progressive censoring

In this paper, we describe how to analyze and design the accelerated life test plans to improve the quality and reliability of the item. We focus on estimating the costs of maintenance service policy because it plays an important role in assisting any manufacturing organization for sale and available equipment and maintenance cost-effective. The step-stress partially accelerated life test is assumed when the lifetime of test units follows Generalized Inverted Exponential distribution under the progressive censoring scheme. The maximum likelihood estimates,Fisher Information matrix, and the asymptotic variance and covariance matrix are obtained. The confidence intervals of the estimators are also obtained. Furthermore, a simulation study is conducted to check the accuracy of the results.

Estimating the Covariance Matrix of the Maximum Likelihood Estimator Under Linear Cluster-Weighted Models

In recent years, the research into cluster-weighted models has been intense. However, estimating the covariance matrix of the maximum likelihood estimator under a cluster-weighted model is still an open issue. Here, an approach is developed in which information-based estimators of such a covariance matrix are obtained from the incomplete data log-likelihood of the multivariate Gaussian linear cluster-weighted model. To this end, analytical expressions for the score vector and Hessian matrix are provided. Three estimators of the asymptotic covariance matrix of the maximum likelihood estimator, based on the score vector and Hessian matrix, are introduced. The performances of these estimators are numerically evaluated using simulated datasets in comparison with a bootstrap-based estimator; their usefulness is illustrated through a study aiming at evaluating the link between tourism flows and attendance at museums and monuments in two Italian regions.

Efficient Estimation for Random Dot Product Graphs via a One-Step Procedure

We propose a one-step procedure to estimate the latent positions in random dot product graphs efficiently. Unlike the classical spectral-based methods, the proposed one-step procedure takes advantage of both the low-rank structure of the expected adjacency matrix and the Bernoulli likelihood information of the sampling model simultaneously. We show that for each vertex, the corresponding row of the one-step estimator (OSE) converges to a multivariate normal distribution after proper scaling and centering up to an orthogonal transformation, with an efficient covariance matrix. The initial estimator for the one-step procedure needs to satisfy the so-called approximate linearization property. The OSE improves the commonly adopted spectral embedding methods in the following sense: Globally for all vertices, it yields an asymptotic sum of squares error no greater than those of the spectral methods, and locally for each vertex, the asymptotic covariance matrix of the corresponding row of the OSE dominates those of the spectral embeddings in spectra. The usefulness of the proposed one-step procedure is demonstrated via numerical examples and the analysis of a real-world Wikipedia graph dataset.