Fig. 14.

(A) Scatterplot of the pairs $\left({\beta }_j,{\widehat{\beta }}_j\right)$ for the dataset from Fig. 3. Here, ${\alpha }_{\star }=1.499$ (red line) and our ProbeFrontier estimate is $\widehat{\alpha }=1.511$ (green line). The estimate is so close that the green line masks the red. (B) True conditional probabilities $\rho \prime \left({\mathit{X}}_i^{\prime }\mathit{\beta }\right)$ (black curve) and corresponding estimated probabilities $\rho \prime \left({\mathit{X}}_i\widehat{\mathit{\beta }}/\widehat{\alpha }\right)$ computed from the debiased MLE (blue points). Observe that the black curve now passes through the center of the blue point cloud. Our predictions are fairly unbiased.