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.
References
[1]
Peelle, R. Peelle's Pertinent Puzzle; Oak Ridge National Laboratory: Oak Ridge, TN, USA, 1987.
[2]
Burr, T.; Kawano, T.; Talou, P.; Hengartner, N.; Pan, P. Defense of the Least Squares Solution to Peelle's Pertinent Puzzle. Algorithms 2011, 4, 28–39.
[3]
International Evaluation of Neutron Cross Section Standards; International Atomic Energy Agency: Vienna, Austria, 2007.
[4]
Zhao, Z.; Perey, R. The Covariance Matrix of Derived Quantities and Their Combination. ORNL/TM-12106; Oak Ridge National Laboratory: Oak Ridge, TN, USA, 1992.
[5]
Chiba, S.; Smith, D. A Suggested Procedure for Resolving an Anomaly in Least-Squares Data Analysis Known as Peelle's Pertinent Puzzle and the General Implications for Nuclear Data Evaluation. ANL/NDM-121; Argonne National Laboratory: Argonne, IL, USA, 1991.
[6]
Chiba, S.; Smith, D. Some Comments on Peelle's Pertinent PuzzleJAERI-M 94-068. Proceeding of Symptoms on Nuclear Data; Japan Atomic Energy Research Institute: Tokai, Japan, 1994; p. 5.
[7]
Chiba, S.; Smith, D. Impacts of Data Transformation on Least-Squares Solutions and Their Significance in Data Analysis and Evaluation. J. Nucl. Sci. Technol. 1994, 31, 770–781.
[8]
Hanson, K.; Kawano, T.; Talou, P. Probabilistic Interpretation of Peelle's Pertinent Puzzle. Proceeding of International Conference of Nuclear Data for Science and Technology, Santa Fe, NM, USA, 26 September–1 October 2004; Haight, R.C., Chadwick, M.B., Kawano, T., Talou, P., Eds.; American Institute of Physics: College Park, MD, USA, 2005.
[9]
Sivia, D. Data Analysis—A Dialogue With The Data. In Advanced Mathematical and Computational Tools in Metrology VII; Ciarlini, P., Filipe, E., Forbes, A.B., Pavese, F., Perruchet, C., Siebert, B.R.L., Eds.; World Scientific Publishing Co.: Singapore, Singapore, 2006; pp. 108–118.
[10]
Jones, C.; Finn, J.; Hengartner, N. Regression with Strongly Correlated Data. J. Multivar. Anal. 2008, 99, 2136–2153.
[11]
Finn, J.; Jones, C.; Hengartner, N. Strong Nonlinear Correlations, Conditional Entropy, and Perfect Estimation, AIP Conference Proceedings 954; American Institute of Physics: College Park, MD, USA, 2007.
[12]
Kawano, T.; Matsunobu, H.; Murata, T.; Zukeran, A.; Nakajima, Y.; Kawai, M.; Iwamoto, O.; Shibata, K.; Nakagawa, T.; Ohsawa, T.; Baba, M.; Yoshida, T. Simultaneous Evaluation of Fission Cross Sections of Uranium and Plutonium Isotopes for JENDL-3.3. J. Nucl. Sci. Technol. 2000, 37, 327–334.
[13]
Kawano, T.; Hanson, K.; Talou, P.; Chadwick, M.; Frankle, S.; Little, R. Evaluation and Propagation of the Pu239 Fission Cross-Section Uncertainties Using a Monte Carlo Technique. Nucl. Sci. Eng. 2006, 153, 1–7.
[14]
Oh, S. Monte Carlo Method for the Estimates in a Model Calculation. Proceedings of Korean Nuclear Society Conference, Yongpyung, Korea, october 2004.
[15]
Oh, S.; Seo, C. Box-Cox Transformation for Resolving Peelle's Pertinent Puzzle in Curve Fitting. PHYSOR-2004; The Physics of Fuel Cycles and Advanced Nuclear Systems: Global Developments: Chicago, IL, USA, 2004.
[16]
Christensen, R. Plane Answers to Complex Questions, The Theory of Linear Models; Springer: New York, NY, USA, 1999; pp. 23–25.
[17]
Burr, T.; Frey, H. Biased Regression: The Case for Cautious Application. Techometrics 2005, 47, 284–296.
[18]
Johnson, R.; Wichern, D. Applied Multivariate Statistical Analysis; Prentice Hall: Upper Saddle River, NJ, USA, 1988.
[19]
R Foundation for Statistical Computing. R: A Language and Environment for Statistical Coputing. 2004. Avalable online: www.R-project.org (accessed on 17 June 2011).
[20]
Schmelling, M. Averaging Correlated Data. Phys. Scr. 1995, 51, 676–679.
[21]
Diggle, P.; Gratton, R. Monte Carlo Methods of Inference for Implicit Statistical Models. J. R. Stat. Soc. B 1984, 46, 193–227.
[22]
Marjoram, P.; Molitor, J.; Plagnol, V.; Tavare, S. Markov Chain Monte Carlo Without Likelihoods. Proc. Nat. Acad. Sci. USA 2003, 100, 15324–15328.
[23]
Plagnol, V.; Tavare, S. Approximate Bayesian Computation and MCMC. Proceedings of Monte Carlo and Quasi-Monte Carlo Methods; National University of Singapore: Singapore, Singapore, 2002; pp. 99–114.
[24]
Blum, M. Approximate Bayesian Computation: A Nonparametric Perspective. J. Am. Stat. Assoc. 2010, 105, 1178–1187.
[25]
Hastie, T.; Tibshirani, R.; Friedman, J. The Elements of Statistical Learning; Springer: New York, NY, USA, 2001.
