Abstract
The sampling distribution of parameter estimators can be summarized by moments, fractiles or quantiles. For nonlinear models, these quantities are often approximated by power series, approximated by transformed systems, or estimated by Monte Carlo sampling. A control variate approach based on a linear approximation of the nonlinear model is introduced here to reduce the Monte Carlo sampling necessary to achieve a given accuracy. The particular linear approximation chosen has several advantages: its moments and other properties are known, it is easy to implement, and there is a correspondence to asymptotic results that permits assessment of control variate effectiveness prior to sampling via measures of nonlinearity. Empirical results for several nonlinear problems are presented.
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References
T.W. Anderson,An Introduction to Multivariate Statistical Analysis (Wiley, New York, 1958).
D.M. Bates and D.G. Watts, Relative curvature measures of nonlinearity, J. Roy. Statist. Soc. B42, 1(1980)1.
E.M.L. Beale, Confidence regions in nonlinear estimation, J. Roy. Statist. Soc. B22, 1(1960)41.
M.J. Box, Bias in nonlinear estimation, J. Roy. Statist. Soc. B33, 1(1971)171.
J.M. Chambers,Computational Methods for Data Analysis (Wiley, New York, 1977).
R.C.H. Cheng, Analysis of simulation experiments under normality assumptions, J. Oper. Res. Soc. 29, 5(1978)493.
G.P.Y. Clarke, Moments of least squares estimators in a nonlinear regression model, J. Roy. Statist. Soc. B42, 2(1980)227.
A.R. Gallant, Nonlinear regression, The American Statistican 29, 2(1975)73.
J.M. Hammersley and D.C. Handscomb,Monte Carlo Methods (Chapman and Hall, London, 1964).
H. Kahn, Use of different Monte Carlo sampling techniques, in:Symposium on Monte Carlo Methods, ed. Meyer (Wiley, New York, 1956) p. 146.
W.J. Kennedy, Jr. and J.E. Gentle,Statistical Computing (Marcel Dekker, New York, 1980).
A.M. Porta Nova, A generalized approach to variance reduction in discrete-event simulation using control variables, unpublished Ph.D. Dissertation, Department of Mechanical Engineering, The University of Texas, Austin, Texas (1985).
G.J.S. Ross, Exact and approximate confidence regions for functions of parameters in nonlinear models, ed. L.C.A. Corsten, COMPSTAT 1978 (1978) 110.
R.Y. Rubinstein and R. Marcus, Efficiency of multivariate control variates in Monte Carlo simulation, Oper. Res. 33, 3(1985)661.
L.R. Shenton and K.O. Bowman,Maximum Likelihood Estimation in Small Samples (Macmillan, New York, 1977).
J.J. Swain, Monte Carlo estimation of sampling distributions arising in nonlinear statistical modeling, unpublished Ph.D. Dissertation, School of Industrial Engineering, Purdue University, West Lafayette, Indiana (1982).
J.J. Swain, Monte Carlo analysis of nonlinear statistical models, III: Efficiency, nonlinearity, and parameter transformations, Technical Report J-84-13, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia (1984).
J.J. Swain and B.W. Schmeiser, Monte Carlo analysis of nonlinear statistical models, I: Theory, Technical Report J-84-11, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia (1984).
J.J. Swain and B.W. Schmeiser, Monte Carlo analysis of nonlinear statistical models, II: Moments and variances, Technical Report J-84-12, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia (1984).
D.E. Tierney, Series expansions with applications to nonlinear models, unpublished Ph.D. Thesis, University of Michigan, Ann Arbor, Michigan (1971).
S. Venkatraman and J.R. Wilson, The efficiency of control variates in multiresponse simulation, Oper. Res. Lett. 5, 1(1986)37.
C.F. Wu, Asymptotic theory of nonlinear least squares estimation, Ann. Statist. 9(1981)501.
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This research was supported in part by the Office of Naval Research under Contract N00014-79-C-0832.
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Swain, J.J., Schmeiser, B.W. Monte carlo estimation of the sampling distribution of nonlinear model parameter estimators. Ann Oper Res 8, 243–256 (1987). https://doi.org/10.1007/BF02187095
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DOI: https://doi.org/10.1007/BF02187095