Elementary Slopes edited version.PDF

In a bivariate data plot, every two points determine an “elementary slope.” For n points with distinct x-values, there are n(n – 1)/2 elementary slopes. These elementary slopes are examined under the two classical regression assumptions: (1) the regressor variable values are fixed and the error is independent and normal, and (2) the data is bivariate normal. For case (1), it is demonstrated that a weighted average of the elementary slopes gives the standard least squares estimate. In case (2), it is shown that the elementary slopes have a rescaled Cauchy distribution; this Cauchy distribution is then used to estimate bivariate normal parameters. Two nonparametric correlation coefficients, Kendall’s τ and the Greatest Deviation correlation coefficient (GD), are used with elementary slopes in regression estimation. Simulations show the robustness of the nonparametric method of estimation using Kendall’s τ and GD.