The University of Montana

Department of Mathematical Sciences

Technical report #1/2004

A Robust Norm Using GDCC

**Rudy Gideon**

University of Montana

Missoula, MT 59812

and

**Carol Ulsafer**

University of Montana

Missoula, MT 59812

**Abstract**

Classically a norm in statistics is essentially the same as a norm in general mathematical analysis. In this work a norm is developed in a completely different, but much more general way, namely via the correlation coefficient.

The author's previous work on location and scale estimates from correlation coefficients will be combined to produce a generalized alternative to the classical norm, called an order norm, as it is based on order statistics. This norm does agree with the classical norm on certain regular data vectors, but this new norm, in contrast to the classical norm, is robust on the unchanged data. Many current robust methods begin with data adjustments to eliminate outlier influence. This method requires no such manipulation.

This paper develops the order norm, shows it is robust for a particular correlation coefficient and that it agrees with the classical norm on certain symmetric data. It illustrates several important properties of the norm and it is shown how to produce a new inner product, a new covariance, and yet another correlation coefficient which leads to further avenues of research. An elaborate example on a classification problem using satellite data is given. The illustrations use the Greatest Deviation correlation coefficient because this nonparametric correlation coefficient makes apparent the generality of the method and gives a robust norm. Any of the correlation coefficients discussed in Gideon (G0, 2000) could be subjected to the same treatment and their particular properties discussed.

**Keywords:** correlation, norm, robust

**AMS Subject Classification:** 62G99, 62G35

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