The Limiting Distribution of the Rank Correlation Coefficient ${R_g}$

Abstract

A new correlation coefficient, ${R_g}$, based on ranks and greatest deviation was defined in Gideon and Hollister (1987). In there the exact distributions were obtained by enumeration for small sample sizes, and by computer simulations for larger sample sizes. In this note, it is shown that the asymptotic distribution of n ^1/2 ${R_g}$ is N(0,1) when the variables are independent and n is the sample size. This limit is derived by restating the definition of ${R_g}$ in terms of a rank measure and then using a limit theorem on set-indexed empirical processes which appears in Pyke (1985). The limiting distribution can be compared to the critical values for large samples given in Figure 2 of Gideon and Hollister (1987). Methods for deriving the limiting distribution under fixed and contiguous alternatives are also described.