Elliptical linear mixed models with a covariate subject to measurement error

• Published in: Statistical papers (Berlin, Germany) Germany), 2020-02, Vol.61 (1), p.31-69
• Main Authors: Borssoi, Joelmir A., Paula, Gilberto A., Galea, Manuel
• Format: Article
• Language: English
• Subjects: Computer simulation; Covariance; Diagnostic systems; Economic models; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Error analysis; Finance; Goodness of fit; Insurance; Iterative methods; Kurtosis; Management; Mathematical models; Mathematics and Statistics; Maximum likelihood estimates; Maximum likelihood method; Monte Carlo simulation; Numerical integration; Numerical methods; Operations Research/Decision Theory; Parameter estimation; Perturbation; Probability Theory and Stochastic Processes; Regular Article; Robustness (mathematics); Statistics; Statistics for Business; Structural hierarchy
• ISSN: 0932-5026; 1613-9798
• DOI: 10.1007/s00362-017-0921-9

In this paper we extend linear mixed models with elliptical errors by adding a covariate subject to measurement error in the linear predictor. The former class is defined appropriately so that the joint marginal distribution of the response and the observed covariate subject to measurement error is also elliptical. Thus, numerical integration methods are not required to obtain the marginal model and the mean and the variance–covariance structures of the hierarchical model are preserved. A kurtosis flexibility is allowed for each joint marginal distribution and since the conditional distributions are also elliptical, the predictions of the random effects as well as of the covariate subject to measurement error may be performed in a similar way of the normal case. A reweighed iterative process based on the maximum likelihood method is derived for obtaining the parameter estimates, which appear to be robust against outlying observations in the sense of the Mahalanobis distance. In order to assess the sample properties of the maximum likelihood parameter estimates as well as their asymptotic standard errors, a simulation study is performed under different parameter settings and error distributions. Goodness-of-fit measures based on the Mahalanobis distance are presented and normal curvatures of local influence are derived under three usual perturbation schemes, which are selected appropriately. Finally, an illustrative example previously analyzed under normal error models is reanalyzed by considering heavy-tailed error models. The diagnostic procedures are applied for comparing the fitted models.