Maximum Likelihood Estimation for a One-Sided Truncated Family of Distributions

For a one-sided truncated family of distributions with an interest parameter θ $$\theta $$ and a truncation parameter γ $$\gamma $$ as a nuisance parameter, we consider the maximum likelihood estimators (MLEs) θ ̂ ML γ $$\hat \theta _{ML}^\gamma $$ and θ ̂ ML $$\hat \theta _{ML}$$ of θ $$\theta $$ for known γ $$\gamma $$ and unknown γ $$\gamma $$ , respectively. In this chapter, the stochastic expansions of θ ̂ ML γ $$\hat \theta _{ML}^\gamma $$ and θ ̂ ML $$\hat \theta _{ML}$$ are derived, and their second order asymptotic variances are obtained. The second order asymptotic loss of a bias-adjusted MLE θ ̂ M L ∗ $$\hat \theta _{ML^*}$$ relative to θ ̂ ML γ $$\hat \theta _{ML}^\gamma $$ 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.