Abstract
Generalized least squares (GLS) for model parameter estimation has a long and successful history dating to its development by Gauss in 1795. Alternatives can outperform GLS in some settings, and alternatives to GLS are sometimes sought when GLS exhibits curious behavior, such as in Peelle’s Pertinent Puzzle (PPP). PPP was described in 1987 in the context of estimating fundamental parameters that arise in nuclear interaction experiments. In PPP, GLS estimates fell outside the range of the data, eliciting concerns that GLS was somehow flawed. These concerns have led to suggested alternatives to GLS estimators. This paper defends GLS in the PPP context, investigates when PPP can occur, illustrates when PPP can be beneficial for parameter estimation, reviews optimality properties of GLS estimators, and gives an example in which PPP does occur.
Highlights
Generalized least squares (GLS) for parameter estimation has a long and successful history dating to its development by Gauss in 1795
Peelle’s Pertinent Puzzle (PPP) was introduced in 1987 in the context of estimating fundamental parameters that arise in nuclear interaction experiments [1]
PPP is described below and when it occurs, the GLS estimate of the parameter is guaranteed to be outside the range of the data, which has elicited concerns that GLS is flawed and has led to suggested alternatives to GLS estimators
Summary
Generalized least squares (GLS) for parameter estimation has a long and successful history dating to its development by Gauss in 1795. The second and third components are independent of the first and of each other, and correspond to 10% random uncertainties in each experimental result This PPP statement is vague, by converting it to something more interpretable, GLS can be applied and the resulting estimate is 0.88 (with an associated standard deviation of 0.22), which is outside the range of the measurements. We are required to obtain the weighted average of those experimental data In this interpretation, the common error (the “fully correlated” component) is understood to be multiplicative, and m1 = 1.5 ± 10% is assumed to mean that the true standard deviation is 0.15 for m1 (and 0.10 for m2 ). In some cases, biased estimators have lower MSE than unbiased estimators because the bias introduced is more than offset by a reduction in variance [8]
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