The University of Montana
Department of Mathematical Sciences

Technical report #1/2004

Elementary Slopes in Simple Linear Regression

Rudy Gideon
University of Montana
Missoula, MT 59812

and

Adele Marie Rothan, CSJ
College of St. Catherine
St. Paul, MN 55105

Abstract

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 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 and GD.

Keywords: bivariate normal, Cauchy distribution, Kendall's , Greatest Deviation, correlation coefficient

AMS Subject Classification: 62G08, 62G35, 62J05

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