Bias correction in a multivariate normal regression model with general parameterization

Abstract:

This paper develops a bias correction scheme for a multivariate normal model under a general parameterization. In the model, the mean vector and the covariance matrix share the same parameters. It includes many impor-tant regression models available in the literature as special cases, such as (non)linear regression, errors-in-variables models, and so forth. Moreover, heteroscedastic situations may also be studied within our framework. We derive a general expression for the second-order biases of maximum likeli-hood estimates of the model parameters and show that it is always possible to obtain the second order bias by means of ordinary weighted lest-squares regressions. We enlighten such general expression with an errors-in-variables model and also conduct some simulations in order to verify the performance of the corrected estimates. The simulation results show that the bias correc-tion scheme yields nearly unbiased estimators. We also present an empirical illustration.

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Multivariate elliptical models with general parameterization

Abstract:

In this paper we introduce a general elliptical multivariate regression model in which the mean vector and the scale matrix have parameters (or/and covariates) in common. This approach unifies several important elliptical models, such as nonlinear regressions, mixed-effects model with nonlinear fixed effects, errors-in-variables models, and so forth. We discuss maximum likelihood estimation of the model parameters and obtain the information matrix, both observed and expected. Additionally, we derived the generalized leverage as well as the normal curvatures of local influence under some perturbation schemes. An empirical application is presented for illustrative purposes.

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A multivariate ultrastructural errors-in-variables model with equation error

Abstract:  This paper deals with asymptotic results on a multivariate ultrastructural errors-in-variables regression model with equation errors. Sufficient conditions for attaining consistent estimators for model parameters are presented. Asymptotic distributions for the line regression estimators are derived. Applications are presented to the elliptical class of distributions with two error assumptions. Model generalizes previous results aimed at univariate scenarios.

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Modified likelihood ratio tests in heteroskedastic multivariate regression models with measurement error

Abstract:

In this paper, we develop a modified version of the likelihood ratio test for multivariate heteroskedastic errors-in-variables regression models. The error terms are allowed to follow a multivariate distribution in the elliptical class of distributions, which has the normal distribution as a special case. We derive the Skovgaard adjusted likelihood ratio statistic, which follows a chi-squared distribution with a high degree of accuracy. We conduct a simulation study and show that the proposed test displays superior finite sample behavior as compared to the standard likelihood ratio test. We illustrate the usefulness of our results in applied settings using a data set from the WHO MONICA Project on cardiovascular disease.

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A heteroscedastic structural errors-in-variables model with equation error.

Abstract:

It is not uncommon with astrophysical and epidemiological data sets that the variances of the observations are accessible from an analytical treatment of the data collection process. Moreover, in a regression model, heteroscedastic measurement errors and equation errors are common situations when modelling such data. This article deals with the limiting distribution of the maximum-likelihood and method-of-moments estimators for the line parameters of the regression model. We use the delta method to achieve it, making it possible to build joint confidence regions and hypothesis testing. This technique produces closed expressions for the asymptotic covariance matrix of those estimators. In the moment approach we do not assign any distribution for the unobservable covariate while with the maximum-likelihood approach, we assume a normal distribution. We also conduct simulation studies of rejection rates for Wald-type statistics in order to verify the test size and power. Practical applications are reported for a data set produced by the Chandra observatory and also from the WHO MONICA Project on cardiovascular disease

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A non-parametric method to estimate the number of clusters

“An important and yet unsolved problem in unsupervised data clustering is how to determine the number of clusters. The proposed slope statistic is a non-parametric and data driven approach for estimating the number of clusters in a dataset. This technique uses the output of any clustering algorithm and identifies the maximum number of groups that breaks down the structure of the dataset. Intensive Monte Carlo simulation studies show that the slope statistic outperforms (for the considered examples) some popular methods that have been proposed in the literature. Applications in graph clustering, in iris and breast cancer datasets are shown.” (Fujita et al., 2014)

Fujita et al. (2014). A non-parametric method to estimate the number of clusters, Computational Statistics and Data Analysis, 73, 27-39.

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