Technical Reports

Technical Reports 2004

Cauchy Regression and Confidence Intervals for the Slope

Rudy Gideon
University of Montana
Missoula, MT 59812

and

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

Abstract

This paper uses computer simulations to verify several features of the Greatest Deviation (GD) nonparametric correlation coefficient. First, its asymptotic distribution is used in a simple linear regression setting where both variables are bivariate. Second, the distribution free property of GD is demonstrated using both the bivariate normal and bivariate Cauchy distributions. Third, the robustness of the method is shown by estimating parameters in the Cauchy case. Fourth, a general geometric method is used to estimate a ratio of scale factors used in the confidence interval. The methods in this paper are an outgrowth of general research on the use of nonparametric correlation coefficients in statistical estimations. The results in this paper are not specific to GD and are appropriate for other rank based correlation coefficients.

Keywords: bivariate normal, bivariate Cauchy, Greatest Deviation correlation coefficient, asymptotic distribution

AMS Subject Classification: 62G08, 62G35, 62G15, 62J05

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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 \(x-\text{values}\), 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 \(\tau\) 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 \(\tau\) and \(GD\).

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

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

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A Robust Norm Using GDCC

Rudy Gideon
University of Montana
Missoula, MT 59812

and

Carol Ulsafer
University of Montana
Missoula, MT 59812

Abstract

Classically a norm in statistics is essentially the same as a norm in general mathematical analysis. In this work a norm is developed in a completely different, but much more general way, namely via the correlation coefficient.

The author's previous work on location and scale estimates from correlation coefficients will be combined to produce a generalized alternative to the classical norm, called an order norm, as it is based on order statistics. This norm does agree with the classical norm on certain regular data vectors, but this new norm, in contrast to the classical norm, is robust on the unchanged data. Many current robust methods begin with data adjustments to eliminate outlier influence. This method requires no such manipulation.

This paper develops the order norm, shows it is robust for a particular correlation coefficient and that it agrees with the classical norm on certain symmetric data. It illustrates several important properties of the norm and it is shown how to produce a new inner product, a new covariance, and yet another correlation coefficient which leads to further avenues of research. An elaborate example on a classification problem using satellite data is given. The illustrations use the Greatest Deviation correlation coefficient because this nonparametric correlation coefficient makes apparent the generality of the method and gives a robust norm. Any of the correlation coefficients discussed in Gideon (G0, 2000) could be subjected to the same treatment and their particular properties discussed.

Keywords: correlation, norm, robust

AMS Subject Classification: 62G99, 62G35

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Location and Scale Estimation with Correlation Coefficients

Rudy Gideon
University of Montana
Missoula, MT 59812

and

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

Abstract

This paper, one in a series on estimation with correlation coefficients, shows how to use any correlation coefficient to produce an estimate of location and scale. Since the normal distribution is so widely used, the method is illustrated using this distribution. Analyzers of normal data are advised to graph a quantile plot to check on the normality assumption before performing their data analysis; Looney and Gulledge (1985) show how to use Pearson's r as a test of normality. This paper shows and recommends that, at this same time, several correlation coefficients can be used to fit a simple linear regression line to the graph and to use the slope and intercept as estimates of standard deviation and location. A robust correlation will produce robust estimates. Tables of mean square error for simulations indicate that the median with this method using a robust correlation coefficient appears to be nearly as efficient as the mean with good data and much better if there are a few possibly errant data points. Hypothesis testing and confidence intervals are illustrated for the scale parameter.

Keywords: simple linear regression, robust estimates, hypothesis testing, confidence intervals

AMS Subject Classification: 62G05, 62G35

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Singularly Perturbed Parabolic Equations with Alternating Boundary Layer Type Solutions

Adelaida B. Vasil'eva
Department of Physics, Moscow State University, Moscow, 119899 Russia
E-mail: abvas@mathabv.phys.msu.su

and

Leonid V. Kalachev
Department of Mathematical Sciences, University of Montana, Missoula, MT 59812, USA
E-mail: kalachev@mso.umt.edu

Abstract

We consider singularly perturbed parabolic equations for which the degenerate equations obtained by setting small parameter to zero are the algebraic equations that have several roots. We study boundary layer type solutions that, as time increases, periodically go through two fairly long lasting stages with extremely fast transitions between these two stages. During one of these stages the solution outside the boundary layer is close to one of the roots of the degenerate (reduced) equation, while during the other stage the solution is close to the other root.

Keywords: singular perturbations, parabolic equations, boundary function method

AMS Subject Classification: 34E10, 35B05, 35B25

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