@inproceedings{
    RoOO01,
    editor = "Dutter, R. and Filzmoser, P. and Gather, U. and Rousseeuw, P. J.",
    author = "Ronkainen, Tommi and Oja, Hannu and Orponen, Pekka",
    publisher = "Springer-Verlag",
    title = "Computation of the Multivariate {O}ja Median",
    url = "http://users.ics.aalto.fi/orponen/publications.html",
    booktitle = "Developments in Robust Statistics: Proceedings of the International Conference on Robust Statistics (ICORS'01, Stift Vorau, Austria, July 2001)",
    address = "Berlin Heidelberg",
    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 $k$-variate data set with $n$ observations our exact and stochastic algorithm have rough complexity estimates of $O(k^2n^k\log n)$ and $O(5^k(1/\varepsilon)^2)$, respectively, where $\varepsilon$ is the radius of confidence $L_{\infty}$-ball.",
    responsibleauthor = "Orponen, Pekka",
    flags = "RAE copy public",
    year = "2003",
    keywords = "multivariate median, multivariate rank, stochastic approximation",
    impactfactor = "D3",
    pages = "344--359"
}