Efficient computation of generalized median estimators

Abstract

The generalized median (GM) estimator is a family of robust estimators that balances the competing demands of statistical efficiency and robustness. By choosing a kernel that is efficient for the parameter, the GM estimator gains robustness by computing the median of the kernel evaluated at all possible subsets from the sample. The GM estimator is often computationally infeasible because the number of subsets can be large for even modest sample sizes. Writing the estimator in terms of the quantile function facilitates an approximation using a sample of all possible subsets. While both sampling with and without replacement are feasible, sampling without replacement is preferred because of the reduction in variance from the sampling fraction. The proposed algorithm uses sequential sampling to compute an approximation within a user-chosen margin of error.