Estimation and hypothesis testing in multivariate linear regression models under non normality

This paper discusses the problem of statistical inference in multivariate linear regression models when the errors involved are non normally distributed. We consider multivariate t-distribution, a fat-tailed distribution, for the errors as alternative to normal distribution. Such non normality is commonly observed in working with many data sets, e.g., financial data that are usually having excess kurtosis. This distribution has a number of applications in many other areas of research as well. We use modified maximum likelihood estimation method that provides the estimator, called modified maximum likelihood estimator (MMLE), in closed form. These estimators are shown to be unbiased, efficient, and robust as compared to the widely used least square estimators (LSEs). Also, the tests based upon MMLEs are found to be more powerful than the similar tests based upon LSEs.     

Robust estimation of the mean vector for high-dimensional data set using robust clustering

The first step in statistical analysis is the parameter estimation. In multivariate analysis, one of the parameters of interest to be estimated is the mean vector. In multivariate statistical analysis, it is usually assumed that the data come from a multivariate normal distribution. In this situation, the maximum likelihood estimator (MLE), that is, the sample mean vector, is the best estimator. However, when outliers exist in the data, the use of sample mean vector will result in poor estimation. So, other estimators which are robust to the existence of outliers should be used. The most popular robust multivariate estimator for estimating the mean vector is S-estimator with desirable properties. However, computing this estimator requires the use of a robust estimate of mean vector as a starting point. Usually minimum volume ellipsoid (MVE) is used as a starting point in computing S-estimator. For high-dimensional data computing, the MVE takes too much time. In some cases, this time is so large that the existing computers cannot perform the computation. In addition to the computation time, for high-dimensional data set the MVE method is not precise. In this paper, a robust starting point for S-estimator based on robust clustering is proposed which could be used for estimating the mean vector of the high-dimensional data. The performance of the proposed estimator in the presence of outliers is studied and the results indicate that the proposed estimator performs precisely and much better than some of the existing robust estimators for high-dimensional data.     

Weighting Method for a Linear Mixed Model

Maximum likelihood is a widely used estimation method in statistics. This method is model dependent and as such is criticized as being non robust. In this article, we consider using weighted likelihood method to make robust inferences for linear mixed models where weights are determined at both the subject level and the observation level. This approach is appropriate for problems where maximum likelihood is the basic fitting technique, but a subset of data points is discrepant with the model. It allows us to reduce the impact of outliers without complicating the basic linear mixed model with normally distributed random effects and errors. The weighted likelihood estimators are shown to be robust and asymptotically normal. Our simulation study demonstrates that the weighted estimates are much better than the unweighted ones when a subset of data points is far away from the rest. Its application to the analysis of deglutition apnea duration in normal swallows shows that the differences between the weighted and unweighted estimates are due to large amount of outliers in the data set.     

Estimation and tests of hypotheses for the initial mean and covariance in the kalman filter model

Kalman filtering techniques are widely used by engineers to recursively estimate random signal parameters which are essentially coefficients in a large-scale time series regression model. These Bayesian estimators depend on the values assumed for the mean and covariance parameters associated with the initial state of the random signal. This paper considers a likelihood approach to estimation and tests of hypotheses involving the critical initial means and covariances. A computationally simple convergent iterative algorithm is used to generate estimators which depend only on standard Kalman filter outputs at each successive stage. Conditions are given under which the maximum likelihood estimators are consistent and asymptotically normal. The procedure is illustrated using a typical large-scale data set involving 10-dimensional signal vectors.     

Inference problems in life testing under multivariate normality

Modified maximum likelihood estimators of the parameters of a multivariate normal distribution are developed when the smallest or largest observations on one of the components are censored. These estimators are used to construct tests for means and correlation coefficients. The robustness of these tests to deviations from normality is investigated.     

Robust estimation in simultaneous equations models

In this paper we review existing work on robust estimation for simultaneous equations models. Then we sketch three strategies for obtaining estimators with a high breakdown point and a controllable efficiency: (a) robustifying three-stage least squares, (b) robustifying the full information maximum likelihood method by minimizing the determinant of a robust covariance matrix of residuals, and (c) generalizing multivariate tau-estimators (Lopuhaä, 1992, Can. J. Statist., 19, 307–321) to these models. They have the same order of computational complexity as high breakdown point multivariate estimators. The latter seems the most promising approach.     

A comparison of some robust,adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.

     

A comparison of some robust, adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.     

Multivariate limited translation empirical Bayes estimators

The paper develops multivariate limited translation empirical Bayes estimators of the normal mean vector which serve as a compromise between the empirical Bayes and the maximum likelihood estimators. These compromise estimators perform better than the regular empirical Bayes estimators, in a frequentist sense, when there is wide departure of an individual observation from the grand average.     

Pairwise likelihood estimation for multivariate mixed Poisson models generated by Gamma intensities

Estimating the parameters of multivariate mixed Poisson models is an important problem in image processing applications, especially for active imaging or astronomy. The classical maximum likelihood approach cannot be used for these models since the corresponding masses cannot be expressed in a simple closed form. This paper studies a maximum pairwise likelihood approach to estimate the parameters of multivariate mixed Poisson models when the mixing distribution is a multivariate Gamma distribution. The consistency and asymptotic normality of this estimator are derived. Simulations conducted on synthetic data illustrate these results and show that the proposed estimator outperforms classical estimators based on the method of moments. An application to change detection in low-flux images is also investigated.     

A Comparison of Methods of Fitting Models to Twin Data

Data on twins are used to infer a genetic component of variance for various quantitative human characteristics. There are several statistical approaches available to analyze twin data. Here we compare three approaches for fitting variance components models to the relationship between height and bi-illiocristal diameter across ages in a sample of male and female Polish twins aged 8–17. Two of the approaches assume a multivariate normal model for the data, with one basing the likelihood on the raw data and the other using the distribution of the sample covariance matrix. The third approach uses a robust modification of the multivariate normal log-likelihood to downweight abnormal observations. The statistical theory underlying the methods is outlined, and the implementation of the methods is discussed.     

New robust estimators for detecting non-random patterns in multivariate control charts: a simulation approach

In the past decade, different robust estimators have been proposed by several researchers to improve the ability to detect non-random patterns such as trend, process mean shift, and outliers in multivariate control charts. However, the use of the sample mean vector and the mean square successive difference matrix in the T ² control chart is sensitive in detecting process mean shift or trend but less sensitive in detecting outliers. On the other hand, the minimum volume ellipsoid (MVE) estimators in the T ² control chart are sensitive in detecting multiple outliers but less sensitive in detecting trend or process mean shift. Therefore, new robust estimators using both merits of the mean square successive difference matrix and the MVE estimators are developed to modify Hotelling's T ² control chart. To compare the detection performance among various control charts, a simulation approach for establishing control limits and calculating signal probabilities is provided as well. Our simulation results show that a multivariate control chart using the new robust estimators can achieve a well-balanced sensitivity in detecting the above-mentioned non-random patterns. Finally, three numerical examples further demonstrate the usefulness of our new robust estimators.     

A robust pairwise likelihood method for incomplete longitudinal binary data arising in clusters

Clustered longitudinal data feature cross‐sectional associations within clusters, serial dependence within subjects, and associations between responses at different time points from different subjects within the same cluster. Generalized estimating equations are often used for inference with data of this sort since they do not require full specification of the response model. When data are incomplete, however, they require data to be missing completely at random unless inverse probability weights are introduced based on a model for the missing data process. The authors propose a robust approach for incomplete clustered longitudinal data using composite likelihood. Specifically, pairwise likelihood methods are described for conducting robust estimation with minimal model assumptions made. The authors also show that the resulting estimates remain valid for a wide variety of missing data problems including missing at random mechanisms and so in such cases there is no need to model the missing data process. In addition to describing the asymptotic properties of the resulting estimators, it is shown that the method performs well empirically through simulation studies for complete and incomplete data. Pairwise likelihood estimators are also compared with estimators obtained from inverse probability weighted alternating logistic regression. An application to data from the Waterloo Smoking Prevention Project is provided for illustration. The Canadian Journal of Statistics 39: 34–51; 2011 © 2010 Statistical Society of Canada     

Robust Estimation for Zero‐Inflated Poisson Regression

Abstract. The zero‐inflated Poisson regression model is a special case of finite mixture models that is useful for count data containing many zeros. Typically, maximum likelihood (ML) estimation is used for fitting such models. However, it is well known that the ML estimator is highly sensitive to the presence of outliers and can become unstable when mixture components are poorly separated. In this paper, we propose an alternative robust estimation approach, robust expectation‐solution (RES) estimation. We compare the RES approach with an existing robust approach, minimum Hellinger distance (MHD) estimation. Simulation results indicate that both methods improve on ML when outliers are present and/or when the mixture components are poorly separated. However, the RES approach is more efficient in all the scenarios we considered. In addition, the RES method is shown to yield consistent and asymptotically normal estimators and, in contrast to MHD, can be applied quite generally.     

Robust estimation of multivariate regression model

This paper studies robust estimation of multivariate regression model using kernel weighted local linear regression. A robust estimation procedure is proposed for estimating the regression function and its partial derivatives. The proposed estimators are jointly asymptotically normal and attain nonparametric optimal convergence rate. One-step approximations to the robust estimators are introduced to reduce computational burden. The one-step local M-estimators are shown to achieve the same efficiency as the fully iterative local M-estimators as long as the initial estimators are good enough. The proposed estimators inherit the excellent edge-effect behavior of the local polynomial methods in the univariate case and at the same time overcome the disadvantages of the local least-squares based smoothers. Simulations are conducted to demonstrate the performance of the proposed estimators. Real data sets are analyzed to illustrate the practical utility of the proposed methodology. This work was supported by the National Natural Science Foundation of China (Grant No. 10471006).     

Shape bias of robust covariance estimators: an empirical study

Detecting outliers in a multivariate point cloud is not trivial, especially when dealing with a sizable fraction of contamination. Over time, it has increasingly been recognized that the safest and most feasible approach to exposing outliers starts by computing a highly robust estimator of location and scatter that can withstand a large proportion of contamination. Many such estimators have been proposed in recent years. We will compare the worst-case bias of several prominent robust multivariate estimators by means of simulation. We also propose a new tool to compare robust estimators on real data sets, and illustrate it.     

Semiparametric estimation and inference for distributional and general treatment effects

Summary.  There is a large literature on methods of analysis for randomized trials with noncompliance which focuses on the effect of treatment on the average outcome. The paper considers evaluating the effect of treatment on the entire distribution and general functions of this effect. For distributional treatment effects, fully non-parametric and fully parametric approaches have been proposed. The fully non-parametric approach could be inefficient but the fully parametric approach is not robust to the violation of distribution assumptions. We develop a semiparametric instrumental variable method based on the empirical likelihood approach. Our method can be applied to general outcomes and general functions of outcome distributions and allows us to predict a subject's latent compliance class on the basis of an observed outcome value in observed assignment and treatment received groups. Asymptotic results for the estimators and likelihood ratio statistic are derived. A simulation study shows that our estimators of various treatment effects are substantially more efficient than the currently used fully non-parametric estimators. The method is illustrated by an analysis of data from a randomized trial of an encouragement intervention to improve adherence to prescribed depression treatments among depressed elderly patients in primary care practices.     

ESTIMATORS BASED ON KENDALL'S TAU IN MULTIVARIATE COPULA MODELS

The estimation of a real‐valued dependence parameter in a multivariate copula model is considered. Rank‐based procedures are often used in this context to guard against possible misspecification of the marginal distributions. A standard approach consists of maximizing the pseudo‐likelihood. Here, we investigate alternative estimators based on the inversion of two multivariate extensions of Kendall's tau developed by Kendall and Babington Smith, and by Joe. The former, which amounts to the average value of tau over all pairs of variables, is often referred to as the coefficient of agreement. Existing results concerning the finite‐ and large‐sample properties of this coefficient are summarized, and new, parallel findings are provided for the multivariate version of tau due to Joe, along with illustrations. The performance of the estimators resulting from the inversion of these two versions of Kendall's tau is compared in the context of copula models through simulations.     

Exact Likelihood Equations for Autoregression Models with Multivariate Elliptically Contoured Distributions

Abstract

The multivariate elliptically contoured distributions provide a viable framework for modeling time-series data. It includes the multivariate normal, power exponential, t, and Cauchy distributions as special cases. For multivariate elliptically contoured autoregressive models, we derive the exact likelihood equations for the model parameters. They are closely related to the Yule-Walker equations and involve simple function of the data. The maximum likelihood estimators are obtained by alternately solving two linear systems and illustrated using the simulation data.     

Minimum distance estimators in extreme value distributions

We define minimum distance estimators for the parameters of the extreme value distribution G_o based on the Cramer-von-Mises distance. These estimators are rather robust and consistent, but asymptotically less efficient than the maximum likelihood estimators which are not robust. A small simulation study for finite sample size show that under G_o the finite efficiency of the minimum distance estimators is rather similar to the maximum likelihood ones.