_Algorithms_**2011**, \(4\), 115-130; doi:10.3390/a4020115  
_algorithms_  
ISSN **1999-4893**  
www.mdpi.com/journal/algorithms  
_Article_  
**Alternatives to the Least Squares Solution to [PERSON]'s Pertinent Puzzle**  
**[PERSON]**\({}^{1,\star}\)**, [PERSON]**\({}^{1}\)**, [PERSON]**\({}^{2}\)**, [PERSON]**\({}^{3}\)**, [PERSON]**\({}^{4}\)  
**and [PERSON]**\({}^{3}\)  
\({}^{1}\) Statistical Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545, USA;  
E-Mail: [EMAIL_ADDRESS]  
\({}^{2}\) Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545, USA;  
E-Mail: [EMAIL_ADDRESS]  
\({}^{3}\) Nuclear and Particle Physics, Los Alamos National Laboratory, Los Alamos, NM 87545, USA;  
E-Mails: [EMAIL_ADDRESS] (T.K.); [EMAIL_ADDRESS] (P.T.)  
\({}^{4}\) Decision Applications, Los Alamos National Laboratory, Los Alamos, NM 87545, USA;  
E-Mail: [EMAIL_ADDRESS]  
\({}^{\star}\) Author to whom correspondence should be addressed; E-Mail: [EMAIL_ADDRESS];  
Tel.: +1-505-665-7865; Fax: +1-505-667-4470.  
_Received: 28 March 2011; in revised form: 1 June 2011 / Accepted: 8 June 2011 / Published: 23 June 2011_  
**Abstract:** Peelle's Pertinent Puzzle (PPP) was described in 1987 in the context of estimating fundamental parameters that arise in nuclear interaction experiments. In PPP, generalized least squares (GLS) parameter estimates fell outside the range of the data, which has raised concerns that GLS is somehow flawed and has led to suggested alternatives to GLS estimators. However, there have been no corresponding performance comparisons among methods, and one suggested approach involving simulated data realizations is statistically incomplete. Here we provide performance comparisons among estimators, introduce approximate Bayesian computation (ABC) using density estimation applied to simulated data realizations to produce an alternative to the incomplete approach, complete the incompletely specified approach, and show that estimation error in the assumed covariance matrix cannot always be ignored.  
**Keywords:** approximate Bayesian computation using density estimation; mean squared error; Peelle's puzzle Introduction  
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 briefly below and in more detail in [2]. When PPP occurs, generalized least squares (GLS) behaves reasonably under the model assumptions [2]. However, in PPP, GLS parameter estimates fall outside the range of the data, which has raised concerns that GLS is somehow flawed and has led to suggested alternatives to GLS estimators.  
Others [1; 3; 4; 5; 6; 7; 8; 9; 12; 13; 14; 15] have investigated alternatives to the GLS approach to PPP, but no numerical performance comparisons to other methods have been reported. And, one ad hoc approach involving simulated data realizations is statistically incomplete [8].  
The following section includes additional background for PPP and the GLS approach. Section 3 discusses the ordinary sample mean as a perhaps naive alternative and shows that estimation error in the assumed covariance matrix cannot always be ignored. Sections 4 and 5 develop an alternative involving a modified form of approximate Bayesian computation (ABC). This ABC-based approach uses density estimation applied to realizations of simulated data that follow an error model appropriate for PPP and having various error distributions. Section 6 provides a second Bayesian alternative resulting from completion of the ad hoc approach by specifying a suitable prior probability density function (pdf). Section 7 is a summary.