Title: Estimating the covariance matrix of the maximum likelihood estimator under linear cluster-weighted models
Authors: Gabriele Soffritti - University of Bologna (Italy) [presenting]
Abstract: Cluster-weighted regression constitutes an approach to regression analysis with random covariates in the presence of unobserved heterogeneity which also allows performing model-based cluster analysis. In recent years the research into this approach has been intense. However, estimating the covariance matrix of the maximum likelihood estimator is still an open issue because the expectation-maximisation algorithm usually employed to estimate parameters of cluster-weighted models does not require deriving an analytical expression for the Hessian matrix; thus, an evaluation of the covariance matrix of the maximum likelihood estimator is generally not available. An approach is developed in which information-based estimators of such a covariance matrix are obtained from the incomplete data log-likelihood of the multivariate Gaussian linear cluster-weighted model. To this end, analytical expressions for the score vector and Hessian matrix are obtained. Three estimators of the asymptotic covariance matrix of the maximum likelihood estimator, based on the score vector and Hessian matrix, are introduced. The performances of these estimators are numerically evaluated using simulated datasets in comparison with a bootstrap-based estimator; their usefulness is illustrated through a study aiming at evaluating the link between tourism flows and attendance at museums and monuments in two Italian regions.