On Adaptive Transformation–Retransformation Estimate of Conditional Spatial Median

Abstract

An affine equivariant modification of the conditional spatial median is proposed and studied. This modification used an adaptive transformation–retransformation procedure based on a data-driven coordinate system. This new estimate of multivariate conditional median improves upon the performance of nonequivariant spatial median especially when there are correlation among the real valued components of the vector of interest as well as when the scales of those components are different. The proposed approach is based on minimizing a loss function equivalent to that in univariate case. We indicate how to compute the proposed estimate and study its asymptotic properties. We also suggest an adaptive procedure to select the optimal data-driven coordinate system. We discuss the performance of our estimator with the help of a finite sample simulation study and illustrate our methodology by a data-set on blood pressure measurements.

Keywords:

• Conditional spatial median
• Equivariance
• Nonpara-metric estimation
• Transformation
• Retransformation

Acknowledgments

The work was supported by the United States Public Service Grant No AG 16996 from the National Institutes of Health. We thank Dr. Charles Rotimi at the National Human Genome Center at Howard University for allowing us to use his blood pressure data for illustration.