_Algorithms_**2011**, \(4\), 28-39; doi:10.3390/a4010028  
_Article_  
**Defense of the Least Squares Solution to Peelle's Pertinent Puzzle**  
**[PERSON]**\({}^{1,\star}\)**, [PERSON]**\({}^{2}\)**, [PERSON]**\({}^{2}\)**, [PERSON]**\({}^{3}\) **and [PERSON]**\({}^{4}\)  
\({}^{1}\) Statistical Sciences, Los Alamos National Laboratory, Los Alamos NM, USA; E-Mail: [EMAIL_ADDRESS]  
\({}^{2}\) Nuclear and Particle Physics, Los Alamos National Laboratory, Los Alamos NM, USA;  
E-Mail: [EMAIL_ADDRESS]  
\({}^{3}\) Decision Applications, Los Alamos National Laboratory, Los Alamos NM, USA;  
E-Mail: [EMAIL_ADDRESS]  
\({}^{4}\) Information Sciences, Los Alamos National Laboratory, Los Alamos NM, USA;  
E-Mail: [EMAIL_ADDRESS]  
\({}^{\star}\) Author to whom correspondence should be addressed; E-Mail: [EMAIL_ADDRESS];  
Tel. : +1-505-664-0513; Fax: +1-505-667-1931. _Received: 6 December 2010; in revised form: 15 January 2011 / Accepted: 7 February 2011 / Published: 15 February 2011_  
**Abstract:** Generalized least squares (GLS) for model parameter estimation has a long and successful history dating to its development by [PERSON] 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. **Keywords:** Peelle's puzzle; mean squared error; measurement error modeling Introduction  
Generalized least squares (GLS) for parameter estimation has a long and successful history dating to its development by [PERSON] in 1795. In some settings, alternatives to GLS can be effective, and are sometimes sought when GLS exhibits curious behavior, such as in [PERSON]'s Pertinent Puzzle (PPP). 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. A GLS estimate lying outside the range of the data causes heartache among nuclear scientists. Therefore, PPP continues to be of theoretical and practical interest. For example, a summary report of International Evaluation of Neutron Cross Section Standards [2] discusses PPP in terms of a standard least-squares procedure. Neutron cross sections are fundamental parameters that describe the probabilities of various neutron interactions. These cross sections are typically estimated from multiple experiments so some type of weighted average estimation scheme is used. The cross section estimates from any two experiments can have shared errors arising for example from using the same measured background, which can lead to covariance structures such as those described below. We quote the original PPP problem proposed by [1] from a report of [PERSON] and [PERSON] [3], \"Suppose we are required to obtain the weighted average of two experimental results for the same physical quantity. The first result is 1.5, and the second result 1.0. The full covariance matrix of those data is believed to be the sum of three components. The first component is fully correlated with standard error 20% of each respective value. The second and third components are independent of the first and of each other, and correspond to 10% random uncertainties in each experimental result.