Functional estimation problem function space selection

The fact is that this is a difficult problem, one which has no commonly accepted solution that applies in all of the many and varied cases where non-linear equations in multiple dimensions need to be fitted. Parameter estimates in such cases are invariably best done in precisely the most non-linear regions of the pertinent variable space (see Chapter 11), where linear approximations do not hold. Several methods have been put forward for both the selection of fitting criteria and the calculation of confidence limits in various instances. One sophisticated method involves the calculation of a matrix of as many rows as there are data points, whose columns are partial derivatives of the fitted function, parameter by parameter. The simplest assumes linear behaviour near the optimum parameters. This variety of choices means that, if confidence limits are to be reported, one must chose one of several available methods of calculating these values. 

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