In statistics, a multivariate median is a location estimate for a multivariate distribution.

Definitions

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An affine invariant median[1] proposed by Hettmansperger and Randles. The estimator has bounded influence function, positive breakdown value, and high efficiency. Compared with other affine equivariant multivariate medians, it has lower computational complexity.

A median has been defined based on spatial sign statistics, called the Oja[2] median, which is an affine equivariant multivariate location estimate with high efficiency, bounded influence, and zero breakdown. Evaluation of the estimate is computationally intensive. Different computational algorithms are discussed in [3] For a k-variate data set with n observations, the computational complexity is   for the exact method, and   for the stochastic algorithm where   is the radius of the L ball.

Affine invariant medians are compared in [4]

References

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  1. ^ A practical affine equivariant multivariate median Thomas P. Hettmansperger and Ronald H. Randles, Biometrika (2002) 89 (4): 851-860. doi: 10.1093/biomet/89.4.851.
  2. ^ Oja, H. (1983). Descriptive statistics for multivariate distributions. Stat. and Prob. Letters, 1, 327–332.
  3. ^ Computation of the multivariate Oja median, T. Ronkainen, H. Oja, P. Orponen, Metrika (year?) (Volume/Issue?) (pages?)
  4. ^ Oja, H. (1999). Affine invariant multivariate sign and rank tests and corresponding estimates: a review. Scand. J. Statist., 26, 319–343.

Niinimaa, A.; Oja, H. (2004). "Multivariate Median". Encyclopedia of Statistical Sciences. New York: John Wiley & Sons, Inc. doi:10.1002/0471667196.ess1107.pub2.