Quasi-Monte Carlo for Highly Structured Generalised Response Models
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Highly structured generalised response models, such as generalised linear mixed models and generalised linear models for time series regression, have become an indispensable vehicle for data analysis and inference in many areas of application. However, their use in practice is hindered by high-dimensional intractable integrals. Quasi-Monte Carlo (QMC) is a dynamic research area in the general problem of high-dimensional numerical integration, although its potential for statistical applications is yet to be fully explored. We survey recent research in QMC, particularly lattice rules, and report on its application to highly structured generalised response models. New challenges for QMC are identified and new methodologies are developed. QMC methods are seen to provide significant improvements compared with ordinary Monte Carlo methods.
KeywordsGeneralised linear mixed models High-dimensional integration Lattice rules Longitudinal data analysis Maximum likelihood Quasi-Monte Carlo Semiparametric regression Serial dependence Time series regression
AMS 2000 Subject Classification62J12 62M10 62M30 65D30 65D32
- P. J. Acklam, “An algorithm for computing the inverse normal cumulative distribution function,” http://home.online.no/∼pjacklam/notes/invnorm/, 2007.
- E. Al-Eid, and J. Pan, “Estimation in generalized linear mixed models using SNTO approximation.” In A. R. Francis, K. M. Matawie, A. Oshlack, and G. K. Smyth (eds.), Proceedings of the 20th International Workshop on Statistical Modelling, pp. 77–84, Sydney, Australia, 2005.Google Scholar
- E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide, Third Edition, SIAM: Philadelphia, 1999.Google Scholar
- BUGS Project, “BUGS: Bayesian Inference Using Gibbs Sampling,” http://www.mrc-bsu.cam.ac.uk/bugs, 2007.
- R. E. Caflisch, W. Morokoff, and A. Owen, “Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension,” Journal of Computational Finance vol. 1 pp. 27–46, 1997.Google Scholar
- D. Clayton, “Generalized linear mixed models.” In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter (eds.), Markov Chain Monte Carlo in Practice, pp. 275–301, Chapman & Hall: London, 1996.Google Scholar
- C. Crainiceanu, D. Ruppert, and M. P. Wand, “Bayesian analysis for penalised spline regression using WinBUGS,” Journal of Statistical Software vol. 14(14), 2005.Google Scholar
- R. A. Davis, W. T. M. Dunsmuir, and Y. Wang, “Modelling time series of count data.” In S. Ghosh (ed.), Asymptotics, Nonparametrics and Time Series, pp. 63–114, Marcel-Dekker: New York, 1999.Google Scholar
- J. Dick, F. Pillichshammer, and B. J. Waterhouse, “The construction of good extensible rank-1 lattices,” Mathematics of Computation, (in press), 2007.Google Scholar
- P. Diggle, K.-L. Liang, and S. Zeger, Analysis of Longitudinal Data, Oxford University Press: Oxford, 1995.Google Scholar
- P. Diggle, P. Heagerty, K.-L. Liang, and S. Zeger, Analysis of Longitudinal Data, Second Edition, Oxford University Press: Oxford, 2002.Google Scholar
- R. Fletcher, Practical Methods of Optimisation, Second Edition, Wiley: Chichester, 1987.Google Scholar
- F. J. Hickernell, and H. S. Hong, “Quasi-Monte Carlo methods and their randomisations.” In R. Chan, Y.-K. Kwok, D. Yao, and Q. Zhang (eds.), Applied Probability, AMS/IP Studies in Advanced Mathematics, vol. 26, pp. 59–77, American Mathematical Society: Providence, 2002.Google Scholar
- P. L’Ecuyer, and C. Lemieux, “Recent advances in randomized quasi-Monte Carlo methods.” In M. Dror, P. L’Ecuyer, and F. Szidarovszki (eds.), Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, pp. 419–474, Kluwer Dordrecht, 2002.Google Scholar
- C. E. McCulloch, and S. R. Searle, Generalized, Linear, and Mixed Models, Wiley: New York, 2000.Google Scholar
- J. Nocedal, and S. J. Wright, Numerical Optimization, Springer, 1999.Google Scholar
- J.-X. Pan, and R. Thompson, “Quasi-Monte Carlo EM algorithm for estimation in generalized linear mixed models.” In R. Payne, and P. Green (eds.), Proceedings in Computational Statistics, pp. 419–424, Physical-Verlag, 1998.Google Scholar
- J. Pan, and R. Thompson, “Quasi-Monte Carlo estimation in generalized linear mixed models.” In A. Biggeri, E. Dreassi, C. Lagazio, M. Marchi (eds.), Proceedings of the 19th International Workshop on Statistical Modelling, pp. 239–243, Firenze University Press: FLorence, 2004.Google Scholar
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, Second Edition, Cambridge University Press, 1995.Google Scholar
- SAS Institute, Inc, http://www.sas.com, 2007.