Identifiability of models for clusterwise linear regression

Identifiability of the parameters is a necessary condition for the existence of consistent estimators. In this paper the identifiability of the parameters of models for data generated by different linear regression distributions with Gaussian errors is investigated. It turns out that such models cause other identifiability problems than do simple Gaussian mixtures. This problem was heretofore ignored; thus there are no satisfying consistency proofs in this area. Three different models are treated: Finite mixture models with random and fixed covariates and a fixed partition model. Counterexamples and sufficient conditions for identifiability are given, including an example for nonidentifiable parameters with an invertible information matrix. The model choice and the interpretation of the parameters are discussed as well as the use of the identifiability concept for fixed partition models. The concept is generalized to "partial identifiability".