Model fitting for multiple variables by minimising the geometric mean deviation

We consider the problem of fitting a linear model for a number of variables but without treating any one of these variables as special, in contrast to regression where one variable is singled out as being a dependent variable. Each of the variables is allowed to have error or natural variability but we do not assume any prior knowledge about the distribution or variance of this variability. The fitting criterion we use is based on the geometric mean of the absolute deviations in each direction. This combines variables using a product rather than a sum and so allows the method to naturally produce units-invariant models; this property is vital for law-like relationships in the natural or social sciences.