Abstract
The Oja median is one of several extensions of the univariate median to the multivariate case. It has many nice properties, but is computationally demanding. In this paper, we first review the properties of the Oja median and compare it to other multivariate medians. Afterwards we discuss four algorithms to compute the Oja median, which are implemented in our R-package OjaNP. Besides these algorithms, the package contains also functions to compute Oja signs, Oja signed ranks, Oja ranks, and the related scatter concepts. To illustrate their use, the corresponding multivariate one- and $C$-sample location tests are implemented.
Highlights
The univariate median is a popular location estimator
The main topic of this paper is to describe the R (R Core Team 2019) package OjaNP (Fischer, Mosler, Möttönen, Nordhausen, Pokotylo, and Vogel 2020), which provides several algorithms for the computation of the Oja median in R and is available from the Comprehensive R Archive Network (CRAN) at https://CRAN.R-project.org/package=OjaNP
The algorithm is driven by the desired final volume of the bounded region, which is the volume of the minimal rectangle containing the region and having edges parallel to the coordinate axes
Summary
The univariate median is a popular location estimator. It is, not straightforward to generalize it to the multivariate case since no generalization is known that retains all properties of the univariate estimator, and different generalizations emphasize different properties of the univariate median. OjaNP: Computing the Oja Median in R extended to the general multivariate case. These and more multidimensional medians are surveyed in Small (1990) and Oja (2013). To demonstrate the application of the Oja median and its sign and rank concepts, Section 2.6 discusses one- and C-sample tests of location. A concept frequently encountered in this paper is affine equivariance. For a matrix-valued scatter statistic S taking on values in Rk×k, affine equivariance is commonly understood as. For a more detailed introduction to affine equivariance, see Oja (2010)
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