A note on ∞manque in variance covariance components models

Stimualted by C.R. Rao's MINQUE J. Focke and G. Dewess introduced the so called r-and ∞ MINQUE. Although they developed a unique charecterization of ∞-MINQUE, they did not give explicite formulas for its computation. The goal this paper is to close this lack and to etend the concept to more general models.     

Econometric disequilibrium models ∗

Four basic strands in the disequilibrium literature are identified. Some examples are discussed and the canonical econometric disequilibrium model and its estimation are dealt with in detail. Specific criticisms of the canonical model,dealing with price and wage rigidity, with the nature of the min condition and the price-adjustment equation, are considered and a variety of modifications is entertained. Tests of the “equilibrium vs. disequilibrium” hypothesis are discussed, as well as several classes of models that may switch between equilibrium and disequilibrium modes. Finally, consideration is given to multimarket disequilibrium models with particular emphasis on the problems of coherence and estimation.     

Testing factor–covariate interaction in rank repeated-measures analysis of covariance models

This article derives the asymptotic properties of rank-based tests for the covariate effects in rank repeated-measures analysis of covariance (ANCOVA) models (Fan and Zhang 2017 Fan, C., and D. Zhang. 2017. Rank repeated measures analysis of covariance. Communications in Statistics - Theory and Methods 46:1158–83.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) employing generalized estimating equation (GEE) techniques. One interested application of the proposed tests is to check the validity of the assumption of homogeneous covariate effects in different levels of the factors. Performance of the proposed tests has been confirmed by simulation studies and illustrated using the famous seizure count data. While the article mainly focuses on interaction tests, the scope of the proposed tests includes testing any contrast of the covariate effect such as the null of no overall covariate effect.     

Precision–imprecision equivalence in a broad class of imprecise hierarchical uncertainty models

Hierarchical models are rather common in uncertainty theory. They arise when there is a ‘correct’ or ‘ideal’ (the so-called first-order) uncertainty model about a phenomenon of interest, but the modeler is uncertain about what it is. The modeler's uncertainty is then called second-order uncertainty. For most of the hierarchical models in the literature, both the first- and the second-order models are precise, i.e., they are based on classical probabilities. In the present paper, I propose a specific hierarchical model that is imprecise at the second level, which means that at this level, lower probabilities are used. No restrictions are imposed on the underlying first-order model: that is allowed to be either precise or imprecise. I argue that this type of hierarchical model generalizes and includes a number of existing uncertainty models, such as imprecise probabilities, Bayesian models, and fuzzy probabilities. The main result of the paper is what I call precision–imprecision equivalence: the implications of the model for decision making and statistical reasoning are the same, whether the underlying first-order model is assumed to be precise or imprecise.     

Positive and negative non-separability for space–time covariance models

Separable spatio-temporal covariance models, defined as the product of purely spatial and purely temporal covariance functions, are often used in practice, but frequently they only represent a convenient assumption. On the other hand, non-separable models are receiving a lot of attention, since they are more flexible to handle empirical covariances showed up in applications. Different forms of non-separability for space–time covariance functions have been recently defined in the literature. In this paper, the notion of positive and negative non-separability is further formalized in order to distinguish between pointwise and uniform non-separability. Various well-known non-separable space–time stationary covariance models are analyzed and classified by using the new definition of non-separability. In particular, wide classes of non-separable spatio-temporal covariance functions, able to capture positive and negative non-separability, are proposed and some examples of these classes are given. General results concerning the non-separability of spatial–temporal covariance functions obtained by a linear combination of spatial–temporal covariance functions and some stability properties are also presented. These results can be helpful to generate as well as to select appropriate covariance models for describing space–time data.     

Heteroskedasticity-consistent covariance matrix estimation:white's estimator and the bootstrap ∗

This paper considers the issue of estimating the covariance matrix of ordinary least squares estimates in a linear regression model when heteroskedasticity is suspected. We perform Monte Carlo simulation on the White estimator, which is commonly used in.

empirical research, and also on some alternatives based on different bootstrapping schemes. Our results reveal that the White estimator can be considerably biased when the sample size is not very large, that bias correction via bootstrap does not work well, and that the weighted bootstrap estimators tend to display smaller biases than the White estimator and its variants, under both homoskedasticity and heteroskedasticity. Our results also reveal that the presence of (potentially) influential observations in the design matrix plays an important role in the finite-sample performance of the heteroskedasticity-consistent estimators.     

Joint mean–covariance model in generalized partially linear varying coefficient models for longitudinal data

In longitudinal data analysis, efficient estimation of regression coefficients requires a correct specification of certain covariance structure, and efficient estimation of covariance matrix requires a correct specification of mean regression model. In this article, we propose a general semiparametric model for the mean and the covariance simultaneously using the modified Cholesky decomposition. A regression spline-based approach within the framework of generalized estimating equations is proposed to estimate the parameters in the mean and the covariance. Under regularity conditions, asymptotic properties of the resulting estimators are established. Extensive simulation is conducted to investigate the performance of the proposed estimator and in the end a real data set is analysed using the proposed approach.     

A monte carlo study of collinearity in linear simultaneous equation models ∗

Although the collinearity issue has been studied in previous simulation studies with a simultaneous system of equations, alternative estimators to circumvent this problem have received little attention. Monte Carlo techniques are used to examine the performance of several estimators under a squared error loss criterion. In particular, this study considers the Vinod–Ullah ridge-type estimators at the first and/or second stage of 2SLS. Ridge regression in the second stage only of 2SLS but not the first stage only, seems to be a practical alternative to 2SLS, especially in situations of strong collinearity. The OLS estimator and the ordinary ridge regression estimator also yield favorable results in situations of moderate to strong collinearity.     

Mixed models and unbalanced data: wherefrom,whereat and whereto?

A brief history of the early years (1820-1947) of random effects models and the estimation of variance components is followed by a personal evaluation of M.L, REML and MINQUE estimation. A method is suggested for combining ML estimator obtained from subsets of a large data set, and comments are made on the need for simulation studies to assess the degree of approximation in using asymptotic properties of ML-type estimators as if they were exact for finite-sized unbalanced data sets.     

Bayesian covariance matrix estimation using a mixture of decomposable graphical models

We present a Bayesian approach to estimating a covariance matrix by using a prior that is a mixture over all decomposable graphs, with the probability of each graph size specified by the user and graphs of equal size assigned equal probability. Most previous approaches assume that all graphs are equally probable. We show empirically that the prior that assigns equal probability over graph sizes outperforms the prior that assigns equal probability over all graphs in more efficiently estimating the covariance matrix. The prior requires knowing the number of decomposable graphs for each graph size and we give a simulation method for estimating these counts. We also present a Markov chain Monte Carlo method for estimating the posterior distribution of the covariance matrix that is much more efficient than current methods. Both the prior and the simulation method to evaluate the prior apply generally to any decomposable graphical model.     

Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data

The article considers Bayesian analysis of hierarchical models for count, binomial and multinomial data using efficient MCMC sampling procedures. To this end, an improved method of auxiliary mixture sampling is proposed. In contrast to previously proposed samplers the method uses a bounded number of latent variables per observation, independent of the intensity of the underlying Poisson process in the case of count data, or of the number of experiments in the case of binomial and multinomial data. The bounded number of latent variables results in a more general error distribution, which is a negative log-Gamma distribution with arbitrary integer shape parameter. The required approximations of these distributions by Gaussian mixtures have been computed. Overall, the improvement leads to a substantial increase in efficiency of auxiliary mixture sampling for highly structured models. The method is illustrated for finite mixtures of generalized linear models and an epidemiological case study.     

Multiple comparisons for an unbalanced a × b design under mixed models with interaction

Consider a two-way factorial experiment involving a “treatment” factor A with fixed effects, a “blocking” factor B with random effects, and interaction effects perhaps non-negligible. The degree of balance required for multiple comparison procedures to be applicable for the comparison of the treatment effects using ordinary least-squares estimates is investigated. For main effects to be estimated independently of MSAB, a sufficient condition is that the design consist of identical blocks, a strong condition of proportional frequencies. Surprisingly, under this condition of proportional frequencies, MSAB does not provide an appropriate variance estimate for inferences on each treatment contrast, even though the statistics F = MSA/MSAB is appropriate for testing equality of the treatment effects. In short, when factor B is random, standard methods of multiple comparisons apply using the interaction mean square MSAB as a variance estimator only when the treatment-block incidences n_n are constant. Nevertheless, for designs with identical blocks, appropriate variance estimates can be identified to allow for conservative or approximate multiple comparisons. This is illustrated for certain treatment balanced designs for comparisons with a control.     

Nonlinear unbiased estimation in linear models †

In the paper the problem of nonlinear unbiased estimation of expectation in linear models is considered. The considerations are restricted to linear plus quadratic estimators with quadratic parts invariant under a group of translations. The one way classification model is considered in detail, for which an explicit formula for the locally best estimators is presented. A numerical evaluation of variances of the best estimators is given for some unbalanced one way classification models and compared with the variance of the ordinary linear estimators.     

Moving average models—time series in m-dimensions

Stochastic models for discrete time series in the time domain are well known but such models lack consideration of spatial dependency I We expand on their work by constructing spatially dependent moving average models. Definitions of order, stationarity, invertibility, autocorrelation function, and spectrum are made as natural extensions of those in zero dimensions and are implemented in the one and two-space dimensional models.     

Adaptive robust estimation in joint mean–covariance regression model for bivariate longitudinal data

The estimation of the covariance matrix is important in the analysis of bivariate longitudinal data. A good estimator for the covariance matrix can improve the efficiency of the estimators of the mean regression coefficients. Furthermore, the covariance estimation itself is also of interest, but it is a challenging job to model the covariance matrix of bivariate longitudinal data due to the complex structure and positive definite constraint. In addition, most of existing approaches are based on the maximum likelihood, which is very sensitive to outliers or heavy-tail error distributions. In this article, an adaptive robust estimation method is proposed for bivariate longitudinal data. Unlike the existing likelihood-based methods, the proposed method can adapt to different error distributions. Specifically, at first, we utilize the modified Cholesky block decomposition to parameterize the covariance matrices. Secondly, we apply the bounded Huber's score function to develop a set of robust generalized estimating equations to estimate the parameters both in the mean and the covariance models simultaneously. A data-driven approach is presented to select the parameter c in the Huber's score function, which can ensure that the proposed method is robust and efficient. A simulation study and a real data analysis are conducted to illustrate the robustness and efficiency of the proposed approach.     

Parameter redundancy in capture–recapture–recovery models

In principle it is possible to use recently derived procedures to determine whether or not all the parameters of particular complex ecological models can be estimated using classical methods of statistical inference. If it is not possible to estimate all the parameters a model is parameter redundant. Furthermore, one can investigate whether derived results hold for such models for all lengths of study, and also how the results might change for specific data sets. In this paper we show how to apply these approaches to entire families of capture–recapture and capture–recapture–recovery models. This results in comprehensive tables, providing the definitive parameter redundancy status for such models. Parameter redundancy can also be caused by the data rather than the model, and how to investigate this is demonstrated through two applications, one to recapture data on dippers, and one to recapture–recovery data on great cormorants.     

Influence diagnostics in Birnbaum–Saunders nonlinear regression models

We consider the issue of assessing influence of observations in the class of Birnbaum–Saunders nonlinear regression models, which is useful in lifetime data analysis. Our results generalize those in Galea et al. [8 Galea, M., Leiva, V. and Paula, G. A. 2004. Influence diagnostics in log-Birnbaum–Saunders regression models. J. Appl. Stat., 31: 1049–1064. [Taylor & Francis Online], [Web of Science ®] [Google Scholar]] which are confined to Birnbaum–Saunders linear regression models. Some influence methods, such as the local influence, total local influence of an individual and generalized leverage are discussed. Additionally, the normal curvatures for studying local influence are derived under some perturbation schemes. We also give an application to a real fatigue data set.     

Testing homogeneity in semiparametric mixture case–control models

Parametric and semiparametric mixture models have been widely used in applications from many areas, and it is often of interest to test the homogeneity in these models. However, hypothesis testing is non standard due to the fact that several regularity conditions do not hold under the null hypothesis. We consider a semiparametric mixture case–control model, in the sense that the density ratio of two distributions is assumed to be of an exponential form, while the baseline density is unspecified. This model was first considered by Qin and Liang (2011 Qin, J., Liang, K.Y. (2011). Hypothesis testing in a mixture case–control model. Biometrics 67(1):182–198.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar], biometrics), and they proposed a modified score statistic for testing homogeneity. In this article, we consider alternative testing procedures based on supremum statistics, which could improve power against certain types of alternatives. We demonstrate the connection and comparison among the proposed and existing approaches. In addition, we provide a unified theoretical justification of the supremum test and other existing test statistics from an empirical likelihood perspective. The finite-sample performance of the supremum test statistics was evaluated in simulation studies.