On Some Functions Involving Mill's Ratio

Abstract

In this note, we prove that, for all (finite) values of $h$, \begin{equation*}\tag{1} \psi(h) = \frac{m_2}{m^2_1} = \frac{1 - h(Z - h)}{(Z - h)^2},\end{equation*} is monotonic increasing, that \begin{equation*}\tag{2} 2m^2_1 - m_2 > 0,\end{equation*} and that \begin{equation*}\tag{3} 1 < \psi(h) < 2,\end{equation*} where $Z$ is the reciprocal of Mill's ratio, \begin{equation*}\tag{4} Z(h) = e^{-h^2/2} \big/ \int^\infty_h e^{-t^2/2} dt,\end{equation*} and where $m_1$ and $m_2$ are respectively the first and second moments of a singly truncated normal distribution about the point of truncation. The function $\psi(h)$ arises in connection with maximum likelihood estimation of population parameters from singly truncated normal samples (cf. for example [1] and references cited therein). The inequality (2) arises in connection with three-moment estimates based on samples of the same type (cf. [2] and [3]).