Estimators with cube root asymptotics are typically the result of M-estimation with non-smooth objective functions. Aside from being inefficient, they are hard to calculate, have intractable limiting distributions, and are unamenable to the bootstrap. Manski's maximum score estimator and irregular histograms receive special attention. We investigate the geometry, algorithmics and robustness properties of Manski's maximum score estimator, a semiparametric estimator of the coefficients in the binary response model. We provide a new exact algorithm for its computation in covariate dimension one and two. This is faster than other algorithms described in the statistical literature. The breakdown point in covariate dimension one is derived, and we make progress towards finding it in higher dimensions. The breakdown points are highly dependent on the underlying data generating mechanism. Irregular histograms on the unit interval are also a major theme of this thesis. These are obtained through the minimisation of the Kullback-Leibler divergence and integrated squared distance. For smooth densities, we derive the limit distributions of the split points estimates for four classes of irregular histograms. Different conditions on the underlying density leads to different rates of convergence, with cube root being the norm. The computational challenges involved in finding these histograms are discussed, and some anomalies associated with them are investigated. Also, it is indicated how one can proceed in order to show consistency of these density estimators. Finally we derive the CIC (cube root information criterion), a cousin of the AIC.