zbMATH — the first resource for mathematics

Numerical integration in logistic-normal models. (English) Zbl 1157.65336
Summary: Marginal maximum likelihood estimation is commonly used to estimate logistic-normal models. In this approach, the contribution of random effects to the likelihood is represented as an intractable integral over their distribution. Thus, numerical methods such as Gauss-Hermite quadrature (GH) are needed. However, as the dimensionality increases, the number of quadrature points becomes rapidly too high. A possible solution can be found among the Quasi-Monte Carlo (QMC) methods, because these techniques yield quite good approximations for high-dimensional integrals with a much lower number of points, chosen for their optimal location. A comparison between three integration methods for logistic-normal models: GH, QMC, and full Monte Carlo integration (MC) is presented. It turns out that, under certain conditions, the QMC and MC method perform better than the GH in terms of accuracy and computing time.

65D30 Numerical integration
65C05 Monte Carlo methods
62-XX Statistics
Full Text: DOI
[1] Abramowitz, M., Stegun, I. (Eds.), 1972. Handbook of Mathematical Functions. Dover Publications Inc., New York. · Zbl 0543.33001
[2] Agresti, A., Categorical data analysis, (2002), Wiley New York · Zbl 1018.62002
[3] Antonov, I.; Saleev, V., An economic method of computing \(\mathit{LP}_\tau\)-sequences, USSR comput. math. math. phys., 19, 252-256, (1979) · Zbl 0432.65006
[4] Booth, J.G.; Hobert, J.H., Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm, J. roy. statist. soc. B, 62, 265-285, (1999) · Zbl 0917.62058
[5] Bratley, P.; Fox, B.L., Algorithm 659 implementing Sobol’s quasirandom sequence generator, ACM trans. math. software, 14, 88-100, (1988) · Zbl 0642.65003
[6] Caflisch, R., Monte Carlo and quasi-Monte Carlo methods, Acta numer., 7, 1-49, (1998) · Zbl 0949.65003
[7] Cools, R., Constructing cubature formulae: the science behind the art, Acta numer., 6, 1-54, (1997) · Zbl 0887.65028
[8] Cools, R., Advances in multidimensional integration, J. comput. appl. math., 149, 1-12, (2002) · Zbl 1013.65019
[9] Crouch, A.; Spiegelman, E., The evaluations of integrals of the form \(\int_{- \infty}^\infty f(t) \exp \left(- t^2\right) \operatorname{d} t\): application to logistic-normal models, J. amer. statist. assoc., 85, 464-469, (1990) · Zbl 0716.65137
[10] Davis, P.; Rabinowitz, P., Methods of numerical integration, (1984), Academic Press Orlando, FL
[11] De Boeck, P.; Wilson, M., Explanatory item response models: A generalized linear and nonlinear approach, (2004), Springer-Verlag New York · Zbl 1098.91002
[12] Fahrmeir, L.; Tutz, G., Multivariate statistical modelling based on generalized linear models, (2001), Springer-Verlag New York · Zbl 0980.62052
[13] Fischer, G., Molenaar, I. (Eds.), 1995. Rasch Models: Foundations and Recent Developments. Springer-Verlag, New York. · Zbl 0815.00010
[14] Halton, J., On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. math., 2, 84-90, (1960) · Zbl 0090.34505
[15] Hickernell, F.; Lemieux, C.; Owen, A., Control variates for quasi-Monte Carlo, Statist. sci., 20, 1-31, (2005) · Zbl 1100.65006
[16] Hosmer, D.; Lemeshow, S., Applied logistic regression, (2000), Wiley New York · Zbl 0967.62045
[17] Jank, W., Quasi-Monte Carlo sampling to improve the efficiency of Monte Carlo EM, Comput. statist. data anal., 48, 685-701, (2005) · Zbl 1429.62021
[18] Judd, K., Numerical methods in economics, (1998), MIT Press Boston · Zbl 0924.65001
[19] Kelley, C.T., Iterative methods for optimization, (1999), SIAM Philadelphia, PA · Zbl 0934.90082
[20] Kocis, L.; Withen, W., Computational investigations of low-discrepancy sequences, ACM trans. math. software, 23, 266-294, (1997) · Zbl 0887.65031
[21] Lemieux, C., L’Ecuyer, P., 2001. On the use of quasi-Monte Carlo methods in computational finance. Available in \(\langle\)http://www.iro.umontreal.ca/\({}^\sim\)lecuyer/myftp/papers/iccs01.pdf⟩.
[22] Lesaffre, E.; Spiessens, B., On the effect of the number of quadrature points in a logistic random-effects model: an example, Appl. statist., 50, 325-335, (2001) · Zbl 1112.62307
[23] Morokoff, W.; Caflisch, R., Quasi-Monte Carlo integration, J. comput. phys., 122, 218-230, (1995) · Zbl 0863.65005
[24] Niederreiter, H., 1992. Random Number Generation and Quasi-Monte Carlo Methods (CBMS-NSF Regional Conference Series in Applied Mathematics, No. 63). · Zbl 0761.65002
[25] Paskov, S., Faster valuation of financial derivatives, J. portfolio management, 22, 113-120, (1995)
[26] Piessens, R.; de Doncker-Kapenga, E.; Uberhuber, C.; Kahaner, D., QUADPACK: A subroutine package for automatic integration, (1983), Springer-Verlag New York · Zbl 0508.65005
[27] R Development Core Team, 2005. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL \(\langle\)http://www.R-project.org⟩.
[28] Rasch, G., Probabilistic models for some intelligence and attainment test, (1960), Danish Institute for Educational Research Copenhagen, Denmark
[29] Rijmen, F.; Briggs, D., Multiple person dimensions and latent item predictors, ()
[30] Rijmen, F.; Tuerlinckx, F.; De Boeck, P.; Kuppens, P., A nonlinear mixed model framework for IRT models, Psychol. methods, 8, 185-205, (2003)
[31] Robert, C.; Casella, G., Monte Carlo statistical methods, (2004), Springer-Verlag New York · Zbl 1096.62003
[32] Shao, J., Mathematical statistics, (2003), Springer-Verlag New York · Zbl 1018.62001
[33] Stroud, A., Approximate calculation of multiple integrals, (1971), Prentice-Hall Englewood Cliffs, NJ · Zbl 0379.65013
[34] Wuertz, D., 2005. fOptions: Financial Software Collection—fOptions. R package version 220.10063. \(\langle\)http://www.rmetrics.org⟩.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.