Relationships Between the UMVU Estimators of the Mean and Median of a Function of a Normal Distribution

Abstract

Let $\eta = f(\mu)$ and $\theta = Ef(Y)$, where $f$ is a monotone function and $Y \sim N(\mu, \sigma^2)$. In this note we use the method of convolution transforms to show that the UMVU estimators of $\eta$ and $\theta$ based on a pair of independent sufficient statistics $T \sim N(\mu, \alpha \sigma^2)$ and $S^2 \sim \sigma^2 \chi^2_{(\nu)}$ are related to each other in a simple manner: the replacement $\alpha$ by $\alpha - 1$ in the expression of the UMVU estimator of $\eta$ gives the corresponding expression of the UMVU estimator of $\theta$. In addition, we show that a similar relationship also exists among the estimators of the variances.