Likelihood-based inference in some partially non-regular exponential families

Maximum likelihood estimation is a standard approach when confronted with the task of finding estimators for parameters when working with data from a known (or at least believed to be known) family of parametric distributions. In problems satisfying regularity conditions maximum likelihood estimators are asymptotically normally distributed and asymptotically efficient. Efficiency means the asymptotic covariance matrix of such estimators is the inverse of the Fisher information matrix. The standard regularity conditions include smoothness of densities in the parameters (three-times differentiability) for all possible observed values;We examine what happens for iid data that comes from a marginal distribution that is regular with the exception that one of the parameters is an unknown lower threshold of detection. Its maximum likelihood estimator is a sample minimum which, necessarily, cannot have an asymptotic normal distribution. (The possible domains of attraction for such parameters are either Weibull, Frechet, or Gumbel.);We demonstrate that in regular exponential families, maximum likelihood estimators of the other parameters of the model are still asymptotically normal with asymptotic covariance matrix equal to the inverse of the Fisher information matrix. We also show that the estimator of the lower threshold is asymptotically exponential and independent of the estimators of the regular parameters asymptotically;For testing purposes the asymptotic independence allows separation of the leading terms in a Taylor expansion of the likelihood ratio test statistic for a point null hypothesis into two independent components, each of which is asymptotically chi2 under the null hypothesis. The component associated with the estimator for the unknown lower threshold has two degrees of freedom. Hence if there are k regular parameters, the resulting test statistic will have an asymptotic distribution that is c2k+2 (in contrast with the c2k+1 null distribution that one would obtain if all parameters were regular);A number of Monte Carlo simulations demonstrate that our asymptotic results are applicable at moderate sample sizes. This holds true regardless whether the lower threshold is a censoring or a truncation point