Maximum Likelihood Estimation of a Natural Parameter for a One-Sided TEF

In the presence of nuisance parameters, the asymptotic loss of the maximum likelihood estimator of an interest parameter was discussed by Akahira and Takeuchi (1982) and Akahira (1986) under suitable regularity conditions from the viewpoint of higher order asymptotics. On the other hand, in statistical estimation in multiparameter cases, the conditional likelihood method is well known as a way of eliminating nuisance parameters (see, e.g., Basu 1997). The consistency, asymptotic normality, and asymptotic efficiency of the MCLE were discussed by Andersen (1970), Huque and Katti (1976), Bar-Lev and Reiser (1983), Bar-Lev (1984), Liang (1984), and others. Further, in higher order asymptotics, asymptotic properties of the MCLE of an interest parameter in the presence of nuisance parameters were also discussed by Cox and Reid (1987) and Ferguson (1992) in the regular case. However, in the nonregular case when the regularity conditions do not necessarily hold, the asymptotic comparison of asymptotically efficient estimators has not been discussed enough in the presence of nuisance parameters in higher order asymptotics yet. For a truncated exponential family of distributions which is regarded as a typical non-regular case, we consider a problem of estimating a natural parameter θ in the presence of a truncation parameter γ as a nuisance parameter. Let θ̂ γ ML and θ̂ML be