The Annals of Mathematical Statistics

On Some Functions Involving Mill's Ratio

D. F. Barrow and A. C. Cohen, Jr.

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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]).

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Ann. Math. Statist., Volume 25, Number 2 (1954), 405-408.

First available in Project Euclid: 28 April 2007

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Barrow, D. F.; Cohen, A. C. On Some Functions Involving Mill's Ratio. Ann. Math. Statist. 25 (1954), no. 2, 405--408. doi:10.1214/aoms/1177728801.

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