LOCAL ASYMPTOTIC THEORY FOR MULTIPLE SOLUTIONS OF LIKELIHOOD EQUATIONS, WITH APPLICATION TO A SINGLE-ION CHANNEL MODEL

This paper investigates local asymptotic theory when there are multipl e solutions of the likelihood equations due to non-identifiability ari sing as a consequence of incomplete information. Local asymptotic resu lts extend in a stochastic way the notion of a Frechet expansion, whic h is then used to prove asymptotic normality of the maximum likelihood estimator under certain conditions. The local theory, which provides simultaneous consistency and asymptotic normality results, is applied to a bivariate exponential model exhibiting non-identifiability. This statistical model arises as an approximation to the distribution of ob servable sojourn-times in the states of a two-state Markov model of a single ion channel and the non-identifiability is a consequence of an inability to record sojourns less than some small detection limit. For each of two possible estimates, confidence intervals are constructed using the asymptotic theory. Although it is not possible to decide bet ween the two estimates on the basis of a single realization, an approp riate solution of the likelihood equations may be found using the addi tional information contained in data based on two different detection limits; this is investigated numerically and by simulation.