Monotonicity in the Noncentrality Parameter of the Ratio of Two Noncentral $t$-Densities

Abstract

Let $p_\nu(t, \delta)$ be the density at $t$ of a noncentral $t$-variable with $\nu$ degrees of freedom and noncentrality parameter $\delta$. It is proved that for any $d > 0$ and fixed $t, p_\nu(t, \delta + d)/p_\nu(t, \delta)$ is a strictly decreasing function of $\delta$.