平特五不中

Non-singularity of the information matrix plays a key role in model identication and the asymptotic theory of statistics. For many statis- tical models, however, this condition seems virtually impossible to verify. An example of such models is a class of mixture models associated with multi- path change-point problems (MCP) which can model longitudinal data with change points. The MCP models are similar in nature to mixture-of-experts models in machine learning. The question then arises as to how often the non- singularity assumption of the information matrix fails to hold. We show that the set of singularities of the information matrix is a nowhere dense set, i.e. geometrically negligible, if the model is identiable and some mild smoothness conditions hold. Under further smoothness conditions we show that the set is also of measure zero, i.e. both geometrically and analytically negligible. In view of these results, we further study a class of semiparametric MCP models, thus paving the way for establishing asymptotic normality of the maximum likelihood estimates (MLE) and statistical inference of the unknown parame- ters in such models. References [1] Asgharian, M. (2014). ON THE SINGULARITIES OF THE INFORMATION MATRIX AND MULTIPATH CHANGE-POINT PROBLEMS. Theory of Probability and its Appli- cations , Vol. 58, No. 4, pp 546-561