Rejoinder to Maximum likelihood estimation of a multi- dimensional log-concave density

Several discussants (Delaigle, Hall, Wellner, Seregin, Chacón, Critchley) ask about other possible shape constraints. Indeed, Seregin and Wellner (2010) have recently shown that a maximum likelihood estimator exists within the class of d-variate densities of the form f = h ◦ g, where h is a known monotone function and g is an unknown convex function. Certain conditions are required on h, but taking h(y) = e recovers log-concavity, while taking h(y) = y 1/r + (with 0 > r > −1/d) yields the larger class of r-concave densities. Questions of uniqueness and computation of the estimate for these larger classes are still open. Of course, such larger classes must still rule out the spiking problem mentioned on p.2 of the paper. Koenker and Mizera (2010) study maximum entropy estimators within these larger classes, while Leng and Jeon propose in their discussion an alternative M -estimation method which again has wide applicability.