On sequential estimation of a normal distribution having equal mean and variance

Mukhopadhyay and Cicconetti \cite{mc2004} derived the Maximum Likelihood Estimator (MLE) and the Uniformly Minimum Variance Unbiased Estimator (UMVUE) of $\theta$ in $N (\theta, \theta)$ and discussed their application to purely sequential and two-stage bounded risk estimation of $\theta$.  In this paper, a much simpler expression is derived for the UMVUE of $\theta$.  Using this expression, a comprehensive investigation is provided for comparing the performances of the sequential estimators based on the MLE and the UMVUE.