Properties of the Geometric Mean Functional Relationship

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Abstract
In this note we present certain hitherto unknown properties of the geometric mean functional relationship (GMFR) approach to linear regression when applied to problems with errors in both variables. Our investigation is motivated by the widespread use of this technique in fisheries research for the estimation of growth rates (Jolicoeur, 1975, Journal of the Fisheries Board of Canada 32, 1491-1494; Sprent and Dolby, 1980, Biometrics 35, 547-550). We show that the GMFR can be interpreted as the estimate that minimizes an error cost functional based on the sum of the triangular areas formed by connecting the measured data points to the estimated line with lines parallel to the coordinate axes. This "least-triangles" approach to GMFR is a useful conceptual device in that it provides an alternative way of looking at GMFR other than as the geometric mean of two least-squares estimates. The GMFR is generally viewed as an ad hoc procedure, whereas the least-triangles approach shows how it can be derived in its own right. It also allows for straightforward derivation of a recursive method for computing the GMFR. This may be of interest in situations where large amounts of data are invloved (see Soh, Evans and Barker, unpublished Technical Report ECE8549, Department of Electrical and Computer Engineering, University of Newcastle, 1985). Finally, we show that there is a close connection between GMFR and the data correlation coefficient thus clarifying certain questions raised by Sprent and Dolby (1980).

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