For a one-sided truncated exponential family of distributions with a truncation parameter γ and a natural parameter θ as a nuisance parameter, the stochastic expansions of the Bayes estimator when θ is known and the Bayes estimator plugging the maximum likelihood estimator (MLE) in θ of when θ is unknown are derived. The second order asymptotic loss of relative to is also obtained through their asymptotic variances. Further, it is shown that and are second order asymptotically equivalent to the bias-adjusted MLEs and when θ is known and when θ is unknown, respectively. Some examples are also given.