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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