Covariance matrix estimation of the maximum likelihood estimator in multivariate clusterwise linear regression

The expectation-maximisation algorithm is employed to perform maximum likelihood estimation in a wide range of situations, including regression analysis based on clusterwise regression models. A disadvantage of using this algorithm is that it is unable to provide an assessment of the sample variability of the maximum likelihood estimator. This inability is a consequence of the fact that the algorithm does not require deriving an analytical expression for the Hessian matrix, thus preventing from a direct evaluation of the asymptotic covariance matrix of the estimator. A solution to this problem when performing linear regression analysis through a multivariate Gaussian clusterwise regression model is developed. Two estimators of the asymptotic covariance matrix of the maximum likelihood estimator are proposed. In practical applications their use makes it possible to avoid resorting to bootstrap techniques and general purpose mathematical optimisers. The performances of these estimators are evaluated in analysing small simulated and real datasets; the obtained results illustrate their usefulness and effectiveness in practical applications. From a theoretical point of view, under suitable conditions, the proposed estimators are shown to be consistent.