102
Views
47
CrossRef citations to date
0
Altmetric
Theory and Method

Nonlinear Experiments: Optimal Design and Inference Based on Likelihood

&
Pages 538-546 | Received 01 Oct 1990, Published online: 27 Feb 2012

References

  • Abdelbasit , K. M. and Plackett , R. L. 1983 . Experimental Design for Binary Data . Journal of the American Statistical Association , 78 : 90 – 98 .
  • Atkinson , A. C. 1982 . Developments in the Design of Experiments . International Statistical Review , 50 : 161 – 177 .
  • Basawa , I. V. and Prakasa Rao , B. L. S. 1980 . Statistical Inference for Stochastic Processes , London : Academic Press .
  • Bates , D. M. 1983 . The Derivative of |X′X| and Its Uses . Technometrics , 25 : 373 – 376 .
  • Bates , D. M. and Watts , D. G. 1980 . Relative Curvature Measures of Nonlinearity . Journal of the Royal Statistical Society , 42 : 1 – 25 . (with discussion), Ser. B
  • Bates , D. M. and Watts , D. G. 1981 . Parameter Transformations for Improved Approximate Confidence Regions in Nonlinear Least Squares . The Annals of Statistics , 9 : 1152 – 1167 .
  • Bates , D. M. and Watts , D. G. 1988 . Nonlinear Regression, Analysis and Its Applications , New York : John Wiley .
  • Behnken , D. W. and Watts , D. G. 1972 . Bayesian Estimation and Design of Experiments for Growth Rates When Sampling From the Poisson Distribution . Biometrics , 28 : 999 – 1009 .
  • Billingsley , P. 1968 . Convergence of Probability Measures , New York : John Wiley .
  • Borkar , V. and Varaiya , P. 1979 . Adaptive Control of Markov Chains I: Finite Parameter Set . IEEE Transactions on Automatic Control , AC-24 : 953 – 958 .
  • Borkar , V. and Varaiya , P. 1982 . Identification and Adaptive Control of Markov Chains . SIAM Journal on Control and Optimization , 20 : 470 – 489 .
  • Box , G. E. P. and Hunter , W. G. “Sequential Design of Experiments for Nonlinear Models,” . Proceedings of the IBM Scientific Computing Symposium on Statistics, October 21–23, 1963 . pp. 113 – 137 .
  • Box , G. E. P. and Lucas , H. L. 1959 . Design of Experiments in Nonlinear Situations . Biometrika , 49 : 77 – 90 .
  • Box , M. J. 1968 . The Occurrence of Replications in Optimal Designs of Experiments to Estimate Parameters in Nonlinear Models . Journal of the Royal Statistical Society , 30 : 290 – 302 . Ser. B
  • Box , M. J. 1970 . Some Experiences With a Nonlinear Experimental Design Criterion . Technometrics , 12 : 569 – 589 .
  • Burkholder , D. L. 1973 . Distribution Function Inequalities for Martingales . The Annals of Probability , 1 : 19 – 42 .
  • Carr , N. L. 1960 . Kinetics of Catalytic Isomerization of n-pentane . Industrial and Engineering Chemistry , 52 : 391 – 396 .
  • Chaloner , K. 1989 . Bayesian Design for Estimating the Turning Point of a Quadratic Regression . Communications in Statistics, Part A–Theory and Methods , 18 : 1385 – 1390 .
  • Chaloner , K. and Larntz , K. 1989 . Optimal Bayesian Design Applied to Logistic Regression Experiments . Journal of Statistical Planning and Inference , 21 : 191 – 208 .
  • Chernoff , H. 1953 . Locally Optimal Designs for Estimating Parameters . Annals of Mathematical Statistics , 30 : 586 – 602 .
  • Chernoff , H. 1975 . “ Approaches in Sequential Design of Experiments ” . In A Survey of Statistical Design and Linear Models , Edited by: Srivastava , J. N. 67 – 90 . New York : North-Holland .
  • Cochran , W. G. 1973 . Experiments for Nonlinear Functions . Journal of the American Statistical Association , 68 : 771 – 781 . (R. A. Fisher Memorial Lecture)
  • Cox , D. R. and Snell , E. J. 1989 . Analysis of Binary Data , London : Chapman and Hall .
  • Draper , N. R. and Hunter , W. G. 1967a . The Use of Prior Distributions in the Design of Experiments for Parameter Estimation in Nonlinear Situations . Biometrika , 54 : 147 – 153 .
  • Draper , N. R. and Hunter , W. G. 1967b . The Use of Prior Distributions in the Design of Experiments for Parameter Estimation in Nonlinear Situations: Multiresponse case . Biometrika , 54 : 662 – 665 .
  • El-Fattah , Y. M. 1981a . Recursive Algorithm for Adaptive Control of Finite Markov Chains . IEEE Transactions on Systems, Man, and Cybernetics , SMC-11 : 135 – 144 .
  • El-Fattah , Y. M. 1981b . Gradient Approach for Recursive Estimation and Control in Finite Markov Chains . Advances in Applied Probability , 13 : 778 – 803 .
  • Fedorov , V. V. 1972 . Theory of Optimal Experiments , New York : Academic Press .
  • Ford , I. 1976 . Optimal Static and Sequential Design: A Critical Review , University of Glasgow, Dept. of Statistics . unpublished Ph.D. dissertation
  • Ford , I. and Silvey , S. D. 1980 . A Sequentially Constructed Design for Estimating a Nonlinear Parametric Function . Biometrika , 67 : 381 – 388 .
  • Ford , I. , Titterington , D. M. and Kitsos , C. P. 1989 . Recent Advances in Nonlinear Experimental Design . Technometrics , 31 : 49 – 60 .
  • Ford , I. , Titterington , D. M. and Wu , C. F. J. 1985 . Inference and Sequential Design . Biometrika , 72 : 545 – 551 .
  • Fries , A. and Bhattacharya , G. K. 1986 . Optimal Design for an Inverse Gaussian Regression Model . Statistics and Probability Letters , 4 : 291 – 294 .
  • Gallant , A. R. 1987 . Nonlinear Statistical Models , New York : John Wiley .
  • Hall , P. and Heyde , C. C. 1980 . Martingale Limit Theory and Its Application , New York : Academic Press .
  • Hamilton , D. C. and Watts , D. G. 1985 . A Quadratic Design Criterion for Precise Estimation in Nonlinear Regression Models . Technometrics , 27 : 241 – 250 .
  • Hamilton , D. C. , Watts , D. G. and Bates , D. M. 1982 . Accounting for Intrinsic Nonlinearity in Nonlinear Regression Parameter Inference Regions . The Annals of Statistics , 10 : 386 – 393 .
  • Hill , P. D. H. 1980 . D-Optimal Designs for Partially Nonlinear Regression Models . Technometrics , 22 : 275 – 276 .
  • Hill , W. J. and Hunter , W. G. 1974 . Design of Experiments for Subsets of Parameters . Technometrics , 16 : 425 – 434 .
  • Hohmann , G. and Jung , W. 1975 . On Sequential and Nonsequential D-Optimal Experiment Design . Biometrisch Zeitschrift , 17 : 329 – 336 .
  • Jenrich , R. J. 1969 . Asymptotic Properties of Nonlinear Least Squares Estimators . Annals of Mathematical Statistics , 40 : 633 – 643 .
  • Khan , M. K. and Yajdi , A. A. 1988 . On D-Optimal Designs for Binary Data . Journal of Statistical Planning and Inference , 18 : 83 – 91 .
  • Khuri , A. I. 1984 . A Note on D-Optimal Designs for Partially Nonlinear Regression Models . Technometrics , 26 : 59 – 61 .
  • Kiefer , J. 1959 . Optimum Experimental Design . Journal of the Royal Statistical Society , 21 : 272 – 319 . (with discussion), Ser. B
  • Kiefer , J. 1961a . Optimum Designs in Regression Problems, II . Annals of Mathematical Statistics , 32 : 298 – 325 .
  • Kiefer , J. “Optimum Experimental Designs V, With Applications to Systematic and Rotatable Designs,” . Proceedings of the Fourth Berkeley Symposium . Vol. 1 , pp. 381 – 405 .
  • Kiefer , J. and Wolfowitz , J. 1959 . Optimum Designs in Regression Problems . Annals of Mathematical Statistics , 30 : 271 – 294 .
  • Kitsos , C. P. 1989 . Fully Sequential Procedures in Nonlinear Design Problems . Computational Statistics and Data Analysis , 8 : 13 – 19 .
  • Kumar , P. R. 1985 . A Survey of Some Results in Stochastic Adaptive Control . SIAM Journal on Control and Optimization , 23 : 329 – 380 .
  • Kumar , P. R. and Becker , A. 1982 . A New Family of Optimal Adaptive Controller for Markov Chains . IEEE Transactions on Automatic Control , AC-27 : 137 – 146 .
  • Kumar , P. R. and Lin , W. 1982 . Optimal Adaptive Controller for Unknown Markov Chains . IEEE Transactions on Automatic Control , AC-27 : 765 – 774 .
  • Lai , T. L. and Wei , C. Z. 1982 . Least Squares Estimates in Stochastic Regression Models With Applications to Identification and Control of Dynamic Systems . The Annals of Statistics , 10 : 154 – 166 .
  • Lauter , E. 1974 . A Method of Designing Experiments for Nonlinear Models . Mathematische Operationsforschung und Statistik , 5 : 697 – 708 .
  • McCormick , W. P. , Mallik , A. K. and Reeves , J. H. 1988 . Strong Consistency of the MLE for Sequential Design Problems . Statistics and Probability Letters , 6 : 441 – 446 .
  • McCullagh , P. 1981 . Discussion of “Randomized Allocation of Treatments in Sequential Experiments,” . Journal of the Royal Statistical Society , 43 : 286 – 287 . by J. A. Bather, Ser. B
  • McCullagh , P. and Nelder , J. 1989 . Generalized Linear Models , London : Chapman and Hall .
  • Minkin , S. 1987 . Optimal Designs for Binary Data . Journal of the American Statistical Association , 82 : 1098 – 1103 .
  • Myers , R. H. , Khuri , A. I. and Carter , W. H. 1989 . Response Surface Methodology: 1966–1988 . Technometrics , 31 : 137 – 157 .
  • Nelder , J. A. and Wedderburn , R. W. M. 1972 . Generalized Linear Models . Journal of the Royal Statistical Society , 135 : 370 – 384 . Ser. A
  • Pazman , A. 1989 . On Information Matrices in Nonlinear Experimental Design . Journal of Statistical Planning and Inference , 21 : 253 – 263 .
  • Prakasa Rao , B. L. S. 1987 . Asymptotic Theory of Statistical Inference , New York : John Wiley .
  • Rasch , D. 1990 . Optimum Experimental Design in Nonlinear Regression . Communications in Statistics, Part A–Theory and Methods , 19 : 4789 – 4806 .
  • Rao , C. R. 1973 . Linear Statistical Inference , New York : John Wiley .
  • Robertazzi , T. G. and Schwartz , S. C. 1989 . An Accelerated Sequential Algorithm for Producing D-Optimal Designs . SIAM Journal on Scientific and Statistical Computing , 10 : 341 – 358 .
  • Sato , M. , Abe , K. and Takeda , H. 1982 . Learning Control of Finite Markov Chains With Unknown Transition Probabilities . IEEE Transactions on Automatic Control , AC-27 : 502 – 505 .
  • Seber , G. A. F. and Wild , C. J. 1989 . Nonlinear Regression , New York : John Wiley .
  • Silvey , S. D. 1980 . Optimal Design , London : Chapman and Hall .
  • Sweeting , T. J. 1980 . Uniform Asymptotic Normality of the Maximum Likelihood Estimator . The Annals of Statistics , 8 : 1375 – 1381 .
  • Sweeting , T. J. 1983 . On Estimator Efficiency in Stochastic Processes . Stochastic Processes and Their Applications , 15 : 93 – 98 .
  • Wald , A. 1943 . On the Efficient Design of Statistical Investigation . Annals of Mathematical Statistics , 14 : 134 – 140 .
  • Wei , C. Z. 1985 . Asymptotic Properties of Least Squares Estimates in Stochastic Regression Models . The Annals of Statistics , 13 : 1498 – 1508 .
  • Woodroofe , M. 1989 . Very Weak Expansions for Sequentially Designed Experiments . The Annals of Statistics , 17 : 1087 – 1102 .
  • Wu , C. F. J. 1981 . Asymptotic Theory of Nonlinear Least Squares Estimation . The Annals of Statistics , 9 : 501 – 513 .
  • Wu , C. F. J. 1985 . Asymptotic Inference From Sequential Design in a Nonlinear Situation . Biometrika , 72 : 553 – 558 .
  • Wu , C. F. J. and Wynn , H. P. 1978 . The Convergence of General Step Length Algorithms for Regular Optimum Design Criteria . The Annals of Statistics , 6 : 1273 – 1285 .
  • Wynn , H. P. 1970 . The Sequential Generation of D-Optimum Experimental Design . Annals of Mathematical Statistics , 41 : 1655 – 1664 .
  • Wynn , H. P. 1972 . Results in the Theory and Construction of D-Optimum Experimental Design . Journal of the Royal Statistical Society , 34 : 133 – 147 . Ser. B
  • Zacks , S. “Problems and Approaches in Design of Experiments for Estimation and Testing in Nonlinear Models,” . Proceedings of the 4th International Symposium on Multivariate Analysis . Edited by: Krishnaiah , P. R. pp. 209 – 223 . Amsterdam : North-Holland .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.