The Utility and General Definition of Correlation Coefficients

Previous attempts at defining other correlation measures mostly tried to generalize the inner product definition used in Pearson’s correlation coefficient. This does not allow for certain useful correlation’s, like the Greatest Deviation, or Gini’s. In this work the idea in Gideon and Hollister (1987) of seeing correlation, as the difference between distance from perfect negative and perfect positive correlation will be used to bring together a general setting. Pearson, Spearman, and Kendall correlation coefficients are then seen as special cases where a linear restriction holds. It will also be seen how to define a wide variety of correlation coefficients. Simple linear regression with these correlations will be discussed in order to illustrate an introduction to statistical estimation with correlation coefficients. The general focus of this paper is simply to outline notation and concepts necessary for using correlation coefficients as estimating functions.