Least squares with non-normal data: estimating experimental variance functions

Abstract

Contrary to popular belief, the method of least squares (LS) does not require that the data have normally distributed (Gaussian) error for its validity. One practically important application of LS fitting that does not involve normal data is the estimation of data variance functions (VFE) from replicate statistics. If the raw data are normal, sampling estimates s² of the variance σ² are χ² distributed. For small degrees of freedom, the χ² distribution is strongly asymmetrical – exponential in the case of three replicates, for example. Monte Carlo computations for linear variance functions demonstrate that with proper weighting, the LS variance-function parameters remain unbiased, minimum-variance estimates of the true quantities. However, the parameters are strongly non-normal – almost exponential for some parameters estimated from s² values derived from three replicates, for example. Similar LS estimates of standard deviation functions from estimated s values have a predictable and correctable bias stemming from the bias inherent in s as an estimator of σ. Because s² and s have uncertainties proportional to their magnitudes, the VFE and SDFE fits require weighting as s⁻⁴ and s⁻², respectively. However, these weights must be evaluated on the calculated functions rather than directly from the sampling estimates. The computation is thus iterative but usually converges in a few cycles, with remaining ‘weighting’ bias sufficiently small as to be of no practical consequence.