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) \(\hat \theta _{ML}^\gamma \) and \(\hat \theta _{ML}\) of \(\theta \) for known \(\gamma \) and unknown \(\gamma \) , respectively. In this chapter, the stochastic expansions of \(\hat \theta _{ML}^\gamma \) and \(\hat \theta _{ML}\) are derived, and their second order asymptotic variances are obtained. The second order asymptotic loss of a bias-adjusted MLE \(\hat \theta _{ML^*}\) relative to \(\hat \theta _{ML}^\gamma \) 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.