Robust estimation of constant and time-varying parameters in nonlinear ordinary differential equation models

References

• Ascher, U.M., and Petzold, L.R. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia: SIAM: Society for Industrial and Applied Mathematics.
   Google Scholar
• Babenko, V.F., Kofanov, V.A., and Pichugov, S.A. (1996), ‘Multivariate Inequalities of Kolmogorov Type and their Applications’, in Multivariate Approximation and Splines, eds. G. Nuraberger, J.W. Schmidt, and G. Walz, Basel: Birkhauser, pp. 1–12.
   Google Scholar
• Burnham, K.P., and Anderson, D.R. (2004), ‘Multimodel Inference: Understanding AIC and BIC in Model Selection’, Sociological Methods & Research, 33(2), 261–304. doi: 10.1177/0049124104268644
   Web of Science ®Google Scholar
• Butcher, J.C. (2008), ‘Numerical Methods for Ordinary Differential Equations. New York: John Wiley and Sons.
   Google Scholar
• Cao, J., Wang, L., and Xu, J. (2011), ‘Robust Estimation for Ordinary Differential Equation Models’, Biometrics, 67(4), 1305–1313. doi: 10.1111/j.1541-0420.2011.01577.x
   PubMed Web of Science ®Google Scholar
• Csendes, T. (1988), ‘Nonlinear Parameter Estimation by Global Optimization-Efficiency and Reliability’, Acta Cybernetica, 8(4), 361–370.
   Google Scholar
• Cui, H. (1997), ‘Asymptotic Normality of M-Estimates in the EV Model’, Systems Science and Mathematical Sciences, 10(3), 225–236.
   Google Scholar
• Grenander, U. (1981), ‘Abstract Inference. New York: Wiley.
   Google Scholar
• Hall, P., and Jones, M. (1990), ‘Adaptive M-Estimation in Nonparametric Regression’, The Annals of Statistics, 18, 1712–1728. doi: 10.1214/aos/1176347874
   Web of Science ®Google Scholar
• Hansen, N. (2009), ‘Benchmarking a BI-Population CMA-ES on the BBOB-2009 Function Testbed’, ACM-GECCO Genetic and Evolutionary Computation Conference, Montreal, pp. 2389–2396.
   Google Scholar
• He, X., Fung, W.K., and Zhu, Z.Y. (2002), ‘Estimation in a Semiparametric Model for Longitudinal Data with Unspecified Dependence Structure’, Biometrika, 89, 579–590. doi: 10.1093/biomet/89.3.579
   Web of Science ®Google Scholar
• Hemker, P. (1972), ‘Numerical Methods for Differential Equations in System Simulation and in Parameter Estimation’, in Analysis and Simulation of Biochemical systems, eds. H.C. Hemker and B. Hess, Amsterdam: North Holland Publishing Company, pp. 59–80.
   Google Scholar
• Huang, J. (1996), ‘Efficient Estimation for the Proportional Hazards Model with Interval Censoring’, The Annals of Statistics, 24, 540–568. doi: 10.1214/aos/1032894452
   Web of Science ®Google Scholar
• Huang, J.Z. (2003), ‘Local Asymptotics for Polynomial Spline Regression’, The Annals of Statistics, 31, 1600–1635. doi: 10.1214/aos/1065705120
   Web of Science ®Google Scholar
• Huang, Y., Liu, D., and Wu, H. (2006), ‘Hierarchical Bayesian Methods for Estimation of Parameters in a Longitudinal HIV Dynamic System’, Biometrics, 62, 413–423. doi: 10.1111/j.1541-0420.2005.00447.x
   PubMed Web of Science ®Google Scholar
• Huber, J., and Ronchetti, E. (2009), Robust Statistics (2nd ed.), New York: Wiley.
   Google Scholar
• Huber, J., and Rossinib, A. (1997), ‘Sieve Estimation for the Proportional-Odds Failure-Time Regression Model with Interval Censoring’, Journal of the American Statistical Association, 92(439), 960–967. doi: 10.1080/01621459.1997.10474050
   Web of Science ®Google Scholar
• Huyer, W., and Neumaier, A. (1999), ‘Global Optimization by Multilevel Coordinate Search’, Journal of Global Optimization, 14(4), 331–355. doi: 10.1023/A:1008382309369
   Web of Science ®Google Scholar
• Jones, D.R., Perttunen, C.D., and Stuckman, B.E. (1993), ‘Lipschitzian Optimization Without the Lipschitz Constant’, Journal of Optimization Theory and Applications, 79(1), 157–181. doi: 10.1007/BF00941892
   Web of Science ®Google Scholar
• Liang, H., Miao, H., and Wu, H. (2010), ‘Estimation of Constant and Time-Varying Dynamic Parameters of HIV Infection in a Nonlinear Differential Equation Model’, Annals of Applied Statistics, 4, 460–483. doi: 10.1214/09-AOAS290
   PubMed Web of Science ®Google Scholar
• Liang, H., and Wu, H. (2008), ‘Parameter Estimation for Differential Equation Models Using a Framework of Measurement Error in Regression Models’, Journal of the American Statistical Association, 103, 1570–1583. doi: 10.1198/016214508000000797
   PubMed Web of Science ®Google Scholar
• Liese, F., and Vajda, I. (1994), ‘Consistency of M-Estimates in General Regression Models’, Journal of Multivariate Analysis, 50(1), 93–114. doi: 10.1006/jmva.1994.1036
   Web of Science ®Google Scholar
• Mattheij, R., and Molenaar, J. (2002), ‘Ordinary Differential Equations in Theory and Practice. Philadelphia: SIAM.
   Google Scholar
• Pollard, D. (1984), Convergence of Stochastic Processes, Berlin: Springer.
   Google Scholar
• Pollard, D. (1985), ‘New Ways to Prove Central Limit Theorems’, Econometric Theory, 1, 295–313. doi: 10.1017/S0266466600011233
   Google Scholar
• Pollard, D. (1990), ‘Empirical Processes Theory and Applications. Hayward: IMS.
   Google Scholar
• Pošik, P., Huyer, W., and Pál, L. (2012), ‘A Comparison of Global Search Algorithms for Continuous Black Box Optimization’, Evolutionary Computation, 20(4), 509–541. doi: 10.1162/EVCO_a_00084
   PubMed Web of Science ®Google Scholar
• Powell, M.J.D. (2008), ‘Developments of NEWUOA for Minimization Without Derivatives’, IMA Journal of Numerical Analysis, 28, 649–664. doi: 10.1093/imanum/drm047
   Web of Science ®Google Scholar
• Ramsay, J.O., Hooker, G., Campbell, D., and Cao, J. (2007), ‘Parameter Estimation for Differential Equations: A Generalized Smoothing Approach (with discussion)’, Journal of the Royal Statistical Society. Series B (Statistical Methodology), 69, 741–796. doi: 10.1111/j.1467-9868.2007.00610.x
   Web of Science ®Google Scholar
• Ramsay, J.O., and Silverman, B.W. (2005), ‘Functional Data Analysis. New York: Springer.
   Google Scholar
• Schumaker, L.L. (1981), ‘Spline Functions. New York: Wiley.
   Google Scholar
• Shen, X. (1997), ‘On Methods of Sieves and Penalization’, The Annals of Statistics, 25, 2555–2591. doi: 10.1214/aos/1030741085
   Web of Science ®Google Scholar
• Van der Vaart, A.W., and Wellner, J.A., Weak Convergence and Empirical Processes: With Applications to Statistics, Springer Series in Statistics, New York, Springer, 1996.
   Google Scholar
• Wang Y.-G., Lin, X., Zhu, M., and Bai, Z. (2007), ‘Robust Estimation Using the Huber Function with a Data-Dependent Tuning Constant’, Journal of Computational and Graphical Statistics, 16(2), 468–481. doi: 10.1198/106186007X180156
   Web of Science ®Google Scholar
• Wellner, J.A., and Zhang, Y. (2007), ‘Two Likelihood-Based Semiparametric Estimation Methods for Panel Count Data with Covariates’, The Annals of Statistics, 35(3), 2106–2142. doi: 10.1214/009053607000000181
   Web of Science ®Google Scholar
• Wu, H., Kumar, A., Miao, H., Holden-Wiltse, J., Mosmann T.R., Livingstone A.M., Belz, G.T., Perelson, A.S., Zand, M.S., and Topham D.J. (2011), ‘Modeling of Influenza-Specific CD8+ T Cells During the Primary Response Indicates that the Spleen is a Major Source of Effectors’, The Journal of Immunology, 187, 4474–4482. doi: 10.4049/jimmunol.1101443
   PubMed Web of Science ®Google Scholar
• Wu, H., Miao, H., Xue, H., Tppham, D., and Zand, M. (2014), ‘Quantifying Immune Response to Influenza Virus Infection via Multivariate Nonlinear ODE Models with Partially Observed State Variables and Time-Varying Parameters’, Statistics in Biosciences, 1867–1764. doi:10.1007/s12561-014-9108-2.
   Google Scholar
• Wu, H., Xue, H., and Kumar, A. (2012), ‘Numerical Discretization-Based Estimation Methods for Ordinary Differential Equation Models Via Penalized Spline Smoothing with Applications in Biomedical Research’, Biometrics, 68(2), 344–352. doi: 10.1111/j.1541-0420.2012.01752.x
   PubMed Web of Science ®Google Scholar
• Xue, H., K.F . Lam, and Li, G. (2004), ‘Sieve Maximum Likelihood Estimator for Semiparametric Regression Models With Current Status Data’, Journal of the American Statistical Association, 99(466), 346–356. doi: 10.1198/016214504000000313
   Web of Science ®Google Scholar
• Xue, H., Miao, H., and Wu, H. (2010), ‘Sieve Estimation of Constant and Time-Varying Coefficients in Nonlinear Ordinary Differential Equation Models by Considering Both Numerical Error and Measurement Error’, The Annals of Statistics, 38(4), 2351–2387. doi: 10.1214/09-AOS784
   PubMed Web of Science ®Google Scholar