Model Fitting Using the Least Volume Criterion
HERTFORDSHIRE UNIV SCHOOL OF BUSINESS (UNITED KINGDOM) DEPT OF STATISTICS ACCOUNTING AND MANAGEMENT SYSTEMS
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Given data on multiple variables we present a method for fitting a function to the data which, unlike conventional regression, treats all the variables on the same basis i.e. there is no distinction between dependent and independent variables. Moreover, all variables are permitted to have error and we do not assume any information is available regarding the errors. The aim is to generate law-like relationships between variables where the data represent quantities arising in the natural and social sciences. Such relationships are referred to as structural or functional models. The method requires that a monotonic relationship exists thus, in the two variable case we do not allow cases where there is zero correlation. Our fitting criterion is simply the sum of the products of the deviations in each dimension and so corresponds to a volume, or more generally a hyper-volume. One important advantage of this criterion is that the fitted models will always he units i.e. scale invariant. We formulate the estimation problem as a fractional programming problem. We demonstrate the method with a numerical example in which we try and uncover the coefficients from a known data-generating model. The data used suffers from multicollinearity and there is preliminary evidence that the least volume method is much more stable against this problem than least squares.
- Numerical Mathematics
- Statistics and Probability