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This paper presents a number of new findings about the canonical change pointestimation problem. The first part studies the estimation of a change point onthe real line in a simple stump model using the robust Huber estimatingfunction which interpolates between the $\ell_1$ (absolute deviation) and$\ell_2$ (least squares) based criteria. While the $\ell_2$ criterion has beenstudied extensively, its robust counterparts and in particular, the $\ell_1$minimization problem have not. We derive the limit distribution of theestimated change point under the Huber estimating function and compare it tothat under the $\ell_2$ criterion. Theoretical and empirical studies indicatethat it is more profitable to use the Huber estimating function (and inparticular, the $\ell_1$ criterion) under heavy tailed errors as it leads tosmaller asymptotic confidence intervals at the usual levels compared to the$\ell_2$ criterion. We also compare the $\ell_1$ and $\ell_2$ approaches in aparallel setting, where one has $m$ independent single change point problemsand the goal is to control the maximal deviation of the estimated change pointsfrom the true values, and establish rigorously that the $\ell_1$ estimationcriterion provides a superior rate of convergence to the $\ell_2$, and thatthis relative advantage is driven by the heaviness of the tail of the errordistribution. Finally, we derive minimax optimal rates for the change planeestimation problem in growing dimensions and demonstrate that Huber estimationattains the optimal rate while the $\ell_2$ scheme produces a rate sub-optimalestimator for heavy tailed errors. In the process of deriving our results, weestablish a number of properties about the minimizers of compound Binomial andcompound Poisson processes which are of independent interest.

On robust learning in the canonical change point problem under heavy tailed errors in finite and growing dimensions