Maximum likelihood estimation for a one-sided truncated family of distributions

Abstract

For a one-sided truncated family of distributions with an interest parameter $$\theta $$ and a truncation parameter $$\gamma $$ as a nuisance parameter, we consider the maximum likelihood estimators (MLEs) $${\hat{\theta }}_\mathrm{ML}^{\gamma} $$ and $${\hat{\theta }}_{\mathrm{ML}}$$ of $$\theta $$ for known $$\gamma $$ and unknown $$\gamma $$ , respectively. In this paper, the stochastic expansions of $${\hat{\theta }}_{\mathrm{ML}}^{\gamma} $$ and $${\hat{\theta }}_\mathrm{ML}$$ are derived, and their second-order asymptotic variances are obtained. The second-order asymptotic loss of a bias-adjusted $${\hat{\theta }}_{\mathrm{ML}^*}$$ relative to $${\hat{\theta }}_{\mathrm{ML}}^{\gamma} $$ is also given. The results are a generalization of those for a one-sided truncated exponential family of distributions. Examples on a one-sided truncated Cauchy distribution, a general truncated exponential family, etc. are also given.