Computing the standard errors of mixture model parameters with EM when classes are well separated (Q1861637)

A finite mixture model is considered with i.i.d. observations \(\xi_i\) with pdf \(\sum_{s=1}^S\pi_s f(\xi_i,\vartheta_s)\), \(f\) being element of the natural exponential family with canonical parameter \(\vartheta\) and \(\pi_s\) are the mixing probabilities. An EM algorithm for \(\pi_s\), \(\vartheta_s\) is considered. It is demonstrated that the complete information matrix for EM estimates is approximately equal to the observed information matrix when the classes are perfectly separated (i.e., the posterior probability for any observation to belong a class is close to either 0 or 1). This fact provides simple estimates for the standard errors of EM estimators in perfectly simulated classes. Results of simulations and real data examples are presented.