The Treatment of Nonidentifiable Models

Abstract

The identifiability of a statistical model, or of the parameters that serve as an index for the model, is one of the pillars on which the classical approach to statistical estimation is based. For parametric classes of distributions represented as {Fθ;θ2Θ}, the parameter q is said to be identifiable if different values of the parameter, say θ1 and θ2, give rise to different distributions Fθ1 and Fθ2 of the observable variable X drawn from a distribution in the class. Without identifiability, a classical estimator bq of the unknown parameter q would necessarily be ambiguous, and thus of little use. The data can only help “identify” an equivalence class in which the parameter appears to reside, but they cannot provide a specific numerical value that would play the role of one’s best guess of the true value of the target parameter. In classical statistical estimation theory, the estimation of a nonidentifiable parameter is viewed, quite simply, as an ill-posed problem.