Location score ($\rho (e)=-f^{\prime }(e)/f(e)$)

The location score is the score function for translating a probability density function $f$. The minus sign comes from differentiating $f(y-\theta )$ in $\theta$. Under regular tail conditions, integration by parts gives $E_f\rho =0$ and $E_f(\epsilon \rho )=1$. These identities determine projections of the score function onto the span of the centered error.