Brenton Clarke

This paper investigates local asymptotic theory when there are multiple solutions of the likelihood equations due to non-identifiability arising as a consequence of incomplete information. Local asymptotic results extend in a stochastic way the notion of a Fréchet expansion, which 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 observable 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 between the two estimates on the basis of a single realization, an appropriate solution of the likelihood equations may be found using the additional information contained in data based on two different detection limits; this is investigated numerically and by simulation.