Identifiability of Finite Linear Regression Mixtures

Identifiability is a necessary condition for the existence of consistent estimates for the parameters of mixture models. In this paper the identifiability of finite mixtures of linear regression models with Normal errors is investigated. Three different models are treated: Mixture models with random and fixed independent variables and a model with fixed partition of the data to the mixture components. Sometimes only parts of the unknown parameter values are of interest. "Partial identifiability" is introduced for this purpose. It turns out that identifiability of finite linear regression mixtures depends on the number of p \Gamma 1-dimensional hyperplanes which one needs to cover the independent variables. Counterexamples and sufficient conditions for identifiability are given for all models. 1 Introduction In general, a stochastic identifiability problem can be explained as follows: Definition 1.1 (Identifiability) Let\Omega be an arbitrary parameter space, P be some space of distr..