Computation of the Multivariate Oja Median

Reference:

Tommi Ronkainen, Hannu Oja, and Pekka Orponen. Computation of the multivariate Oja median. In R. Dutter, P. Filzmoser, U. Gather, and P. J. Rousseeuw, editors, Developments in Robust Statistics: Proceedings of the International Conference on Robust Statistics (ICORS'01, Stift Vorau, Austria, July 2001), pages 344–359, Berlin Heidelberg, 2003. Springer-Verlag.

Abstract:

The multivariate Oja (1983) median 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 Oja sample median, and the confidence region shrinks to the sample median with the increasing number of the sampled hyperplanes. Regular grids and and the grid given by the data points are used in the construction. Computation times of different algorithms are discussed and compared. For a -variate data set with observations our exact and stochastic algorithm have rough complexity estimates of and , respectively, where is the radius of confidence -ball.

Keywords:

multivariate median, multivariate rank, stochastic approximation

Suggested BibTeX entry:

@inproceedings{RoOO01,
    address = {Berlin Heidelberg},
    author = {Tommi Ronkainen and Hannu Oja and Pekka Orponen},
    booktitle = {Developments in Robust Statistics: Proceedings of the International Conference on Robust Statistics (ICORS'01, Stift Vorau, Austria, July 2001)},
    editor = {R. Dutter and P. Filzmoser and U. Gather and P. J. Rousseeuw},
    pages = {344--359},
    publisher = {Springer-Verlag},
    title = {Computation of the Multivariate {O}ja Median},
    year = {2003},
}