Variance-covariance Matrix Research Articles

Finite-sample performance of the robust variance estimator in the presence of missing data

Theoretically, the maximum likelihood estimator has the sandwich-type asymptotic variance-covariance matrix under model misspecification. Its empirical estimator, that is called the robust variance estimator, is consistent. Thus, the estimator is asymptotically valid even under model misspecification. In practice, the robust variance estimator is used for computation of standard errors in longitudinal data analysis. Recently, Golden et al. (2019 Econometrics, 7, 1-27) showed that the maximum likelihood estimator retains a sandwich-type asymptotic variance-covariance matrix in the presence of missing data even when the missing-data mechanism is missing not at random. Although they revealed the asymptotic validity of the robust variance estimator in the simultaneous presence of both model misspecification and missing data, its finite-sample performance did not be investigated. In this article, we evaluated the finite-sample performance via simulation studies and clarify its small-sample problems. In addition, we illustrated the robust variance estimator using longitudinal CD4 count data from a randomized double-blind study.

Bias correction for multiple covariate analysis using empirical bayesian estimation in mixed-effects models for longitudinal data

The naïve empirical Bayes method has been widely used as an ad hoc tool in fitting linear mixed-effect models, which is much computationally efficient than the maximum likelihood estimation method. However, the shrinkage effect of the empirical Bayes method causes bias in the estimates of the fixed effects. Bias-correction has been proposed for the mixed-effects model when only one covariate is present. In this paper, we derive the shrinkage factor of the empirical Bayes predictors of the random effects and the variance-covariance matrix of the corrected estimates when the model has more than one covariate. The empirical Bayes estimates and test statistics are then corrected using the derived factor. Theoretical derivations, simulation studies and a real data application demonstrate the validity of the proposed method in that the corrected estimates are unbiased and the corrected tests have correct p-values.

Screening: From tornado diagrams to effective dimensions

Popular sensitivity analysis techniques such as Tornado Diagrams or the Morris method are based on one-at-a-time input variations (local main effects, henceforth). We evidence a link between local main effects of screening methods and global sensitivity measures. The link allows analysts to obtain insights on interactions on the local and global scales simultaneously. We evidence a mirroring effect according to which a local main effect calculated from the base case to the sensitivity case is equal in magnitude to the local total effect in the opposite direction, and their difference yields an overall local interaction index. We then obtain the exact relationship between Morris sensitivity measures and Sobol’ total indices and discuss its implications on the choice of the sampling scheme. We show that the non-diagonal elements of the variance-covariance matrix of main effects computed at randomized locations yield Liu and Owen’s global interaction indices. Asymptotic results are derived with particular reference to confidence intervals for the mean dimension. We illustrate findings and insights through numerical experiments on two well known simulators.

Repeated-Measures Analysis in the Context of Heteroscedastic Error Terms with Factors Having Both Fixed and Random Levels

The design and analysis of experiments which involve factors each consisting of both fixed and random levels fit into linear mixed models. The assumed linear mixed-model design matrix takes either a full-rank or less-than-full-rank form. The complexity of the data structures of such experiments falls in the model-selection and parameter-estimation process. The fundamental consideration in the estimation process of linear models is the special case in which elements of the error vector are assumed equal and uncorrelated. However, different assumptions on the structure of the variance–covariance matrix of error vector in the estimation of parameters of a linear mixed model may be considered. We conceptualise a repeated-measures design with multiple between-subjects factors, in which each of these factors has both fixed and random levels. We focus on the construction of linear mixed-effects models, the estimation of variance components, and hypothesis testing in which the default covariance structure of homoscedastic error terms is not appropriate. We illustrate the proposed approach using longitudinal data fitted to a three-factor linear mixed-effects model. The novelty of this approach lies in the exploration of the fixed and random levels of the same factor and in the subsequent interaction effects of the fixed levels. In addition, we assess the differences between levels of the same factor and determine the proportion of the total variation accounted for by the random levels of the same factor.

Efficient estimation of a Cox model when integrating the subgroup incidence rate information

Incidence rates for diseases are widely used in the field of medical research because they lead to clear and simple physical and clinical interpretations. In this study, we propose an efficient estimation method that incorporates auxiliary subgroup information related to the incidence rate into the estimation of the Cox proportional hazard model. The results show that utilizing the incidence rate information improves the efficiency of the estimation of regression parameters based on the double empirical likelihood method compared to that for conventional models that do not incorporation such information. We show that estimators of regression parameters asymptotically follow a multivariate normal distribution with a variance-covariance matrix that can be consistently estimated. Simulation results indicate that the proposed estimators significantly increase efficiency. Finally, an example of the effects of type 2 diabetes on stroke is applied to demonstrate the proposed method.

Learning target class eigen subspace (LTC-ES) via eigen knowledge grid

In one-class classification (OCC) tasks, only the target class (class-of-interest (CoI)) samples are well defined during training, whereas the other class samples are totally absent. In OCC algorithms, the high dimensional data adds computational overhead apart from its intrinsic property of curse of dimensionality. For target class learning, conventional dimensionality reduction (DR) techniques are not suitable due to negligence of the unique statistical properties of CoI samples. In this context, the present research proposes a novel target class guided DR technique to extract the eigen knowledge grid that contains the most promising eigenvectors of variance-covariance matrix of CoI samples. In this process the lower and higher eigenvalued eigenvectors are rejected via statistical analysis because the high variance may split the target class itself, whereas the lower variance do not contribute significant information. Furthermore, the identified eigen knowledge grid is utilized to transform high dimensional samples to the lower dimensional eigen subspace. The proposed approach is named as learning target class eigen subspace (LTS-ES) that ensures strong separation of the target class from other classes. To show the effectiveness of transformed lower dimensional eigen subspace, one class support vector machine (OCSVM) has been experimented on wide variety of benchmark datasets in presence of: original feature space, transformed features obtained via eigenvectors of approximately 80%-90% cumulative variance, transformed features obtained via knowledge grid and transformed features obtained via eigenvectors of approximately 50% cumulative variance. Finally, a new performance measure parameter called stability factor is introduced to validate the robustness of the proposed approach.

Connectedness between G10 currencies: Searching for the causal structure

AbstractThis paper presents a new approach for modelling the connectedness between asset returns. We adapt the measure of Diebold and Yilmaz, which is based on the forecast error variance decomposition of a VAR model. However, their connectedness measure hinges on critical assumptions with regard to the variance–covariance matrix of the error terms. We propose to use a more agnostic empirical approach, based on a machine learning algorithm, to identify the contemporaneous structure. In a Monte Carlo study, we compare the different connectedness measures and discuss their advantages and disadvantages. In an empirical application we analyse the connectedness between the G10 currencies. Our results suggest that the US dollar as well as the Norwegian krone are the most independent currencies in our sample. By contrast, the Swiss franc and New Zealand dollar have a negligible impact on other currencies. Moreover, a cluster analysis suggests that the currencies can be divided into three groups, which we classify as: commodity currencies, European currencies and safe haven/carry trade financing currencies.

Were the socio-economic determinants of municipalities relevant to the increment of COVID-19 related deaths in Brazil in 2020?

The COVID-19 pandemic in Brazil has been showing a pattern of distribution of related deaths associated with individual socioeconomic status (SES). However, little is known about the role of SES in the distribution of the mortality rate in different population, from an ecological perspective. The objective of this study was to evaluate the role of socioeconomic factors in the distribution of the COVID-19-related mortality rate among Brazilian municipalities in 2020. We conducted a retrospective, cross-sectional, observational, population-wide, and ecological study, using data of COVID-19-related deaths from the Influenza Epidemiological Surveillance Information System database (SIVEP-Gripe) and SES from the Social Vulnerability Index (SVI), the Human Development Index (HDI), the Geographic Index of the Socioeconomic Context and Social Studies (GeoSES), and 2010 Demographic Census (IBGE/Brazil). We computed crude, age- and sex-standardized, and the latter offset by the time of exposure to the epidemic mortality rates. To determine socioeconomic factors associated with mortality rates we used log-linear models with state codes as a random effect and Haversine variance-covariance matrix. 191,528 deaths were related to COVID-19 and distributed in 4,928 (88.55%) Brazilian municipalities. Whatever the socioeconomic indexes used, the R2 were very small to explain SMRT. Consistent across all socioeconomic indexes used, high-income, more educated, and well infrastructure municipalities generally had higher mortality rates. Excluding the effect of demographic structure and pandemic timing from mortality rates, the contribution of SES to explain differences in COVID-19-related mortality rates among municipalities in Brazil became very low. The impact of SES on COVID-19-related mortality may vary across levels of aggregation. Urban infrastructure, which includes mobility structures, more complex economic activities and connections, may have influenced the average municipal death rate.

An Alternative Estimation Method for Time-Varying Parameter Models

A multivariate, non-Bayesian, regression-based, or feasible generalized least squares (GLS)-based approach is proposed to estimate time-varying VAR parameter models. Although it has been known that the Kalman-smoothed estimate can be alternatively estimated using GLS for univariate models, we assess the accuracy of the feasible GLS estimator compared with commonly used Bayesian estimators. Unlike the maximum likelihood estimator often used together with the Kalman filter, it is shown that the possibility of the pile-up problem occurring is negligible. In addition, this approach enables us to deal with stochastic volatility models, models with a time-dependent variance–covariance matrix, and models with non-Gaussian errors that allow us to deal with abrupt changes or structural breaks in time-varying parameters.

A quantitative approach to the choice of number of samples for percentile estimation in bootstrap and visual predictive check analyses

Understanding the uncertainty in parameter estimates or in derived secondary variables is important in all data analysis activities. In pharmacometrics, this is often done based on the standard errors from the variance–covariance matrix of the estimates. Confidence intervals derived in this way are by definition symmetrical, which may lead to implausible outcomes, and will require translation to generate uncertainties in derived variables. An often‐used alternative is numerical percentile estimation by, for example, nonparametric bootstraps to circumvent these issues. Visual predictive checks (VPCs), which is a commonly used model diagnostic tool in pharmacometric analyses, also rely on the estimation of percentiles through numerical approaches. Given the cost in terms of run times and processing times for these methods, it is important to consider the trade‐off between the number of bootstrap samples or simulated data sets in the VPCs, to the increase in precision related to a large number of bootstrap samples or simulated data sets. The objective with this tutorial is to provide a quantitative framework for assessing the precision in estimated percentile limits in bootstrap and visual predictive checks analyses to facilitate an informed choice of confidence interval width, number of bootstrap samples/simulated data sets, and required level of precision.

Nuclear data uncertainty propagation applied to the versatile test reactor conceptual design

The Versatile Test Reactor (VTR) currently under development is a 300 MWth sodium-cooled fast reactor (SFR) fueled with ternary metal alloy fuel, which aims to accelerate the testing of advanced nuclear fuels, materials, instrumentation, and sensors in high flux environments that are necessary to license the next generation of advanced reactor concepts. To support the VTR design process, uncertainties associated with the nuclear data has been propagated through the reactor core neutronics calculation to global parameters of interest, such as the core multiplication factor, kinetic parameters, and various reactivity feedback coefficients, following the sensitivity based uncertainty propagation approach. By folding the sensitivity coefficients, separately computed by the generalized perturbation theory code PERSENT and Monte Carlo code Serpent 2, with the variance–covariance matrices from COMMARA-2.0, we obtain the reaction-wise, isotope-wise, and overall uncertainties for each response of interest due to nuclear data uncertainty. With Serpent 2, the statistical error of the uncertainty is obtained by propagating the statistical error of the sensitivity coefficients through the same process using a newly developed uncertainty propagation method. From both codes, the overall top uncertainty contributors are found to be the cross section of Fe-56 elastic scattering, Na-23 elastic scattering, and U-238 inelastic scattering. The large contributions of the Fe-56 elastic scattering cross sections to global parameters are due to its relatively large relative uncertainty of 5–10% in nuclear data and the large volume of Fe-containing reflector assemblies in the fairly compact VTR core design. Both codes agreed well for the overall uncertainty estimates of all responses of interest, except the delayed neutron fraction, prompt neutron generation time, and the coolant density feedback coefficient, where Serpent 2 yielded a much larger value than PERSENT due to the large statistical error of sensitivity coefficients. The calculated uncertainties are also compared to those associated with other SFR cores. Another outcome of this study is a variance–covariance matrix of reactivity coefficients, which can be used in the subsequent uncertainty propagation to the system level to investigate the impact of identified uncertainties on system responses in the safety analysis.

Regularizing ill-posed problem of single-epoch precise GNSS positioning using an iterative procedure

Abstract This paper analyses the regularization of an ill-conditioned mathematical model in a single-epoch precise GNSS positioning. The regularization parameter (RP) is selected as a parameter that minimizes the criterion of the Mean Squared Error (MSE) function. The crucial for RP estimation is to ensure stable initial least-squares (LS) estimates to replace the unknown quadratic matrix of actual values with the LS covariance matrix. For this purpose, two different data models are proposed, and two research scenarios are formed. Two regularized LS estimations are tested against the non-regularized LS approach. The first one is the classic regularization of LS estimation. In turn, the second one is its iterative counterpart. For the LS estimator of iterative regularization, regularized bias is significantly lower while the overall accuracy is improved in the sense of MSE. The regularized variance-covariance matrix of better precision can mitigate the impact of regularized bias on integer least-squares (ILS) estimation up to some extent. Therefore, iterative LS regularization is well-designed for single-epoch integer ambiguity resolution (AR). Nevertheless, the performance of the ILS estimator is studied in the context of the probability of correct integer AR in the presence of regularized bias.

Inferences for Stress-Strength Reliability Model in the Presence of Partially Accelerated Life Test to Its Strength Variable.

We focus on estimating the stress-strength reliability model when the strength variable is subjected to the step-stress partially accelerated life test. Based on the assumption that both stress and strength random variables follow Weibull distribution with a common first shape parameter, the inferences for this reliability system are constructed. The maximum likelihood, two parametric bootstraps, and Bayes estimates are obtained. Moreover, approximate confidence intervals, asymptotic variance-covariance matrix, and highest posterior density credible intervals are derived. A simulation study and application to real-life data are conducted to compare the proposed estimation methods developed here and also check the accuracy of the results.

A Robust Comparison Powers of Four Multivariate Analysis of Variance Tests

This paper compared the powers of four test statistics of Multivariate Analysis of Variance and test statistics include Lawley-Hotelling ,Pillai's trace, Roy’s largest root and Wilks' lambda. The R Statistics was employed to simulate the data used to compare the four test statistics under the Multivariate Gamma and Multivariate Normal distributions. The sample sizes utilized were 10, 20, 30, 40, 100, 200, 300, 400, 600, 700, 800 and 1000); Number of variables (p = 2, 3, and 4) equal and unequal samples and variance co-variance matrix were also used. The comparison were done at two levels of significance (α = 0.01 and 0.05) using power of the test. The results obtained indicated that the Roy’s largest Root test statistic is better than all other test statistic considered when p = g = 2 because it has the least powers. The result of the analysis further showed that when p = g = 3 and p = g = 4 the Wilks' lambda proved better than all other test statistics both for small and large sample sizes. The results equally showed that when the data are multivariate normal and Gamma with g =2 and p = 2 the power of the four test statistics from best to least is Roy’s largest root, followed Lawley’s trace = Pillae’s trace and the least is Wilks’ Lambda at significant levels of 0.01 and 0.05 for equal and unequal samples. More so, when data are multivariate normal as well as gamma and p= g = 3 and p=g=4, the power of the four statistics ranked from best to least is Wilks’ Lambda , Pillae’s trace = Lawley’s trace and Roy’s largest root respectively . From the findings, it is evidently obvious that Roy’s largest root should be applied when p=g=2 while Wilks’ Lambda be used when p=g=3 and p=g=4. This study will help researchers to plan studies with controlled probabilities of detecting a meaningful effect thereby giving conclusive results with maximum efficiency.

Unidentifiability of errors-in-variables models with rank deficiency from measurements

Although errors-in-variables (EIV) models have been substantially investigated, rank deficient EIV models have not been properly addressed. This paper proves the following major results: (i) given a full EIV model that the elements of its coefficient matrix are continuous random variables with a positive definite variance–covariance matrix, such an EIV model cannot be rank deficient with probability one, indicating that the non-uniqueness condition of total least squares solutions by Golub and van Loan [1] and van Huffel and Vandewalle [2, 3] cannot be true with probability one; (ii) even if a full EIV model is theoretically rank deficient, its rank deficiency cannot be identifiable from measurements with probability one. A full EIV model with theoretical rank deficiency can almost always be incorrectly identified to be an estimable model with probability one, due to the randomness of the coefficient matrix; and (iii) if an EIV model is rank deficient, it must be a partial EIV model, though a partial EIV model is not necessarily rank deficient. Further, if a partial EIV model is rank deficient, we may not numerically obtain its exact rank. These theoretical results have a practical engineering implication that solutions to a rank deficient EIV model from random measurements can be totally incorrect without the notice of an engineer, since there exist an infinite number of mathematically equivalent solutions to such an EIV model. We propose two methods to solve a partial rank deficient EIV model for a unique solution by imposing extra subjective information.

Spectrally Consistent Mean Dynamic Topography by Combining Mean Sea Surface and Global Geopotential Model Through a Least Squares-Based Approach

The filtering procedure is usually mandatory for modeling mean dynamic topography (MDT) when a geodetic approach based on the Mean Sea Surface (MSS) and the Global Geopotential Model (GGM) is used. This is due to the inconsistent spectral contents between MSS and GGM. However, traditional isotropic filtering algorithms (e.g., Gaussian filter) consider neither the MDT locations nor their azimuth when smoothing the signal within the filtering radius. Hence, the isotropic filtering will attenuate the MDT signal near the current and filter the current signal into the surrounding ocean, which may lead to signal contamination and distortion. In this study, we set up a least squares-based (LS) approach to model MDT signal from the altimeter-derived MSS and geoid height using spherical harmonics from GGMs, where MDT is parameterized by Lagrange Basis Functions (LBFs). The design matrix is segmentally established, considering the error information of GGM in various spectral bands. Numerical experiments in the Gulf Stream show that applications of full error variance-covariance matrix or only diagonal error variance of GGM may have marginal effects on the MDT modeling. The MDT computed from this LS-based approach using the latest releases of Gravity Field and Steady-state Ocean Circulation Explorer (GOCE) geoid models, i.e., GO_CONS_GCF_2_DIR_R6 and Gravity Observation Combination 06s model (GOCO06s), have the best agreement with the comparison data, especially near the current region. Deduced geostrophic velocities based on the MDT solutions show that the LS-based approach recovers the current signal better than the Gaussian filtering by 1.8 cm/s. Estimated error map illustrates that errors are more concentrated near the coastal region.

Unified M-estimation of matrix exponential spatial dynamic panel specification

In this paper, a unified M-estimation method in Yang (2018) is extended to the matrix exponential spatial dynamic panel specification (MESDPS) with fixed effects in short panels. Similar to the STLE model which includes the spatial lag effect, the space-time effect and the spatial error effect in Yang (2018), the quasi-maximum likelihood (QML) estimation for MESDPS also has the initial condition specification problem. The initial-condition free M-estimator in this paper solves this problem and is proved to be consistent and asymptotically normal. An outer product of martingale difference (OPMD) estimator for the variance-covariance (VC) matrix of the M-estimator is also derived and proved to be consistent. The finite sample property of the M-estimator is studied through an extensive Monte Carlo study. The method is applied to US outward FDI data to show its validity.

Analysis of genotype-by-environment interactions in a maize mapping population.

Genotype-by-environment interactions are a significant challenge for crop breeding as well as being important for understanding the genetic basis of environmental adaptation. In this study, we analyzed genotype-by-environment interactions in a maize multiparent advanced generation intercross population grown across 5 environments. We found that genotype-by-environment interactions contributed as much as genotypic effects to the variation in some agronomically important traits. To understand how genetic correlations between traits change across environments, we estimated the genetic variance–covariance matrix in each environment. Changes in genetic covariances between traits across environments were common, even among traits that show low genotype-by-environment variance. We also performed a genome-wide association study to identify markers associated with genotype-by-environment interactions but found only a small number of significantly associated markers, possibly due to the highly polygenic nature of genotype-by-environment interactions in this population.

Mean Equality Tests for High-Dimensional and Higher-Order Data with k-Self Similar Compound Symmetry Covariance Structure

An extension of the D2 test statistic to test the equality of mean for high-dimensional and k-th order array-variate data using k-self similar compound symmetry (k-SSCS) covariance structure is derived. The k-th order data appear in many scientific fields including agriculture, medical, environmental and engineering applications. We discuss the property of this k-SSCS covariance structure, namely, the property of Jordan algebra. We formally show that our D2 test statistic for k-th order data is an extension or the generalization of the D2 test statistic for second-order data and for third-order data, respectively. We also derive the D2 test statistic for third-order data and illustrate its application using a medical dataset from a clinical trial study of the eye disease glaucoma. The new test statistic is very efficient for high-dimensional data where the estimation of unstructured variance-covariance matrix is not feasible due to small sample size.

Quantifying the effect of probability model misspecification in seismic collapse risk assessment

One of the main steps in probabilistic seismic collapse risk assessment is estimating the fragility function parameters. The maximum likelihood estimation (MLE) approach, which is widely used for this purpose, contains the underlying assumption that the likelihood function is known to follow a specified parametric probability distribution. However, this assumed distribution may not always be consistent with the “true” probability distribution of the collapse data. This paper implements the Information matrix equivalence theorem to identify the presence of model misspecification i.e., if the assumed collapse probability distribution is, in fact, the “true” one. In the presence of model misspecification, the fragility parameter estimates continue to be asymptotically normally distributed but the variance–covariance matrix is no longer equal to the inverse of the Fisher’s Information matrix. To increase the robustness of the variance–covariance matrix, the Huber–White sandwich estimator is implemented. Using collapse data from eight woodframe buildings, the effect of model misspecification on fragility parameter estimates and collapse rate is quantified. For the considered building cases, the parameter estimation uncertainty in the collapse risk did not increase when the “sandwich” estimator was used compared to when probability model misspecification was not considered (i.e., using MLE). The proposed framework should be used to further investigate the issue of probability model misspecification as it relates to fragility parameter estimation since only a single construction type (woodframe buildings) and limit state (collapse) was considered in the current study.