Special Case of the Distribution of the Median

Abstract

Let $t$ be the translation parameter of a process $X(t), -\infty < t < \infty$. The likelihood ratio of the process $X(t)$ at $t$ against $t = 0$ can be written as $\exp\lbrack W(t) - \frac{1}{2}|t|\rbrack, -\infty < t < \infty$, where $W(t)$ is a standard Wiener process. For the absolute error-loss function the best invariant estimator of the translation parameter is the median of the posterior distribution. The distribution of the median for the posterior distribution is obtained, when the prior distribution for $t$ is the Lebesgue measure on the real line.