On the Variance Covariance Matrix of the Maximum Likelihood Estimator of a Discrete Mixture

The estimation of models involving discrete mixtures is a common practice in econometrics, for example to account for unobserved heterogeneity. However, the literature is relatively uninformative about the measurement of the precision of the parameters. This note provides an analytical expression for the observed information matrix in terms of the gradient and hessian of the latent model when the number of components of the discrete mixture is known. This in turn allows for the estimation of the variance covariance matrix of the ML estimator of the parameters. I discuss further two possible applications of the result: the acceleration of the EM algorithm and the specification testing with the information matrix test.