A Strictly conditional approach in estimation theory

Abstract

Abstract Let X 1, X 2, ... be vector-valued random variables and let the distribution of X i depend on two parameters θ and τ i where θ has the same value for all i while the value of τ i changes with i. Following Neyman & Scott [8] we shall denote θ a structural parameter and τ i an incidental parameter. It was shown by Neyman & Scott that a simultaneous maximum likelihood estimation of θ and τ1 τ2, ... , on may lead to an inconsistent estimation of θ. Methods for obtaining consistent estimates for a structural parameter in the presence of infinitely many incidental parameters have been suggested by Neyman & Scott [8], Kiefer & Wolfowitz [4] and Andersen [2]. In Andersen [2] the consistent solution was obtained as a conditional maximum likelihood estimator given minimal sufficient statistics T i = T(X i); i = 1, 2, ... for the incidental parameters. As shown in Andersen [1] a minimal sufficient statistic for τ1 in the presence of θ will in general depend on θ. The conditional maximum likelihood method re...