MVUE for the Mean With One Observation: Normal with Same Mean and Variance.

Based on one observation X from a N(θ, θ) distribution where θ(> 0) is an unknown parameter, this article provides an exact expression of the minimum variance unbiased estimator of θ based on the complete and sufficient statistic ∣X∣. Even though N(θ, θ) belongs to a one-parameter exponential family, I discuss some small surprises and intricacies that are worth noting. [ABSTRACT FROM AUTHOR]