Computation of the Multivariate Oja Median

Summary

The multivariate Oja median (Oja, 1983) is an affine equivariant multivariate location estimate with high efficiency. This estimate has a bounded influence function but zero breakdown. The computation of the estimate appears to be highly intensive. We consider different, exact and stochastic, algorithms for the calculation of the value of the estimate. In the stochastic algorithms, the gradient of the objective function, the rank function, is estimated by sampling observation. hyperplanes. The estimated rank function with its estimated accuracy then yields a confidence region for the true sample Oja median, and the confidence region shrinks to the sample median with the increasing number of the sampled hyperplanes. Regular grids and the grid given by the data points are used in the construction. Computation times of different algorithms are discussed and compared. For a k-variate data set with n observations our exact and stochastic algorithms have rough time complexity estimates of O(k ² n ^k log n) and O(5^k (1/ε)²), respectively, where ε is the radius of confidence L_∞-ball.