Location and Scale Estimation with Correlation Coefficients

Abstract

This article shows how to use any correlation coefficient to produce an estimate of location and scale. It is part of a broader system, called a correlation estimation system (CES), that uses correlation coefficients as the starting point for estimations. The method is illustrated using the well-known normal distribution. This article shows that any correlation coefficient can be used to fit a simple linear regression line to bivariate data and then the slope and intercept are estimates of standard deviation and location. Because a robust correlation will produce robust estimates, this CES can be recommended as a tool for everyday data analysis. 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 errant data points. Hypothesis testing and confidence intervals are discussed for the scale parameter; both normal and Cauchy distributions are covered.

Keywords:

• Confidence intervals
• Hypothesis testing
• Robust estimates
• Simple linear regression

Mathematics Subject Classification:

• Primary 62G05, 62G08
• Secondary 62G10, 62G30, 62G35

Acknowledgments

R. A. Gideon is indebted to many people for support over the years; please see http://www.umt.edu/math/People/Gideon.html. In particular, he thanks the following Masters’ students for their interest and help on ideas developed in this article: J. Bruder, L. C. Lee, H. Li, and M. Thiel. Their curiosity and enthusiasm have kept this research alive. Also, Carol Ulsafer gave valuable editorial assistance.

Much of the work on CES is available on the website: www.umt.edu/math/ People/Gideon.html. Some of the references will refer to particular articles posted at this website.

Notes

Entries for s, s′, AD, MAD, d _F , s _bi are from Iglewicz [1983],pp. 410, 424.