Georgia Institute of Technology College of Sciences

I will review recent progress concerning nonparametric estimation of log-concave densities and related families in $R^1$ and $R^d$. In the case of $R^1$, I will present limit theory for the estimators at fixed points at which the population density has a non-zero second derivative and for the resulting natural mode estimator under a corresponding hypothesis. In the case of $R^d$ with $d\ge 2$ will briefly discuss some recent progress and sketch a variety of open problems.