Convexity of probit weights

We demonstrate that the probit weight function is U-shaped on R, i.e., it is strictly decreasing on (−∞,0), strictly increasing on [0,∞), and strictly convex on R. Knowledge of the shape of the probit weight function can be useful in several contexts. For instance, it can resolve any confusion that may arise from a result in the classic paper of Sampford (1953). The shape of the probit weight function can also be used to justify why the computation of the probit maximum likelihood estimator (MLE) -in fact, in general, the computation of all two-step estimators whenever the first step involves estimating a probit model -may fail in the presence of outliers or unbounded parameter spaces.