Nonlinear Experiments: Optimal Design and Inference Based on Likelihood

Abstract Nonlinear experiments involve response and regressors that are connected through a nonlinear regression-type structure. Examples of nonlinear models include standard nonlinear regression, logistic regression, probit regression. Poisson regression, gamma regression, inverse Gaussian regression, and so on. The Fisher information associated with a nonlinear experiment is typically a complex nonlinear function of the unknown parameter of interest. As a result, we face an awkward situation. Designing an efficient experiment will require knowledge of the parameter, but the purpose of the experiment is to generate data to yield parameter estimates! Our principal objective here is to investigate proper designing of nonlinear experiments that will let us construct efficient estimates of parameters. We focus our attention on a very general nonlinear setup that includes many models commonly encountered in practice. The experiments considered have two fundamental stages: a static design in the initial stage,...

[1]  A. Wald.  On the Efficient Design of Statistical Investigations , 1943 .

[2]  H. Chernoff.  Locally Optimal Designs for Estimating Parameters , 1953 .

[3]  M. O. Glasgow.  Note on the Factorial Moments of the Distribution of Locally Maximal Elements in a Random Sample , 1959 .

[4]  H. L. Lucas,et al.  DESIGN OF EXPERIMENTS IN NON-LINEAR SITUATIONS , 1959 .

[5]  J. Kiefer.  Optimum Experimental Designs , 1959 .

[6]  J. Kiefer,et al.  Optimum Designs in Regression Problems , 1959 .

[7]  N. L. Carr.  Kinetics of Catalytic Isomerization of n-Pentane , 1960 .

[8]  J. Kiefer.  Optimum Experimental Designs V, with Applications to Systematic and Rotatable Designs , 1961 .

[9]  George E. P. Box,et al.  SEQUENTIAL DESIGN OF EXPERIMENTS FOR NONLINEAR MODELS. , 1963 .

[10]  W. G. Hunter,et al.  The use of prior distributions in the design of experiments for parameter estimation in non-linear situations. , 1967, Biometrika.

[11]  W. G. Hunter,et al.  The use of prior distributions in the design of experiments for parameter estimation in non-linear situations: multiresponse case. , 1967, Biometrika.

[12]  M. J. Box.  The Occurrence of Replications in Optimal Designs of Experiments to Estimate Parameters in Non‐Linear Models , 1968 .

[13]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[14]  R. Jennrich.  Asymptotic Properties of Non-Linear Least Squares Estimators , 1969 .

[15]  H. Wynn.  The Sequential Generation of $D$-Optimum Experimental Designs , 1970 .

[16]  M. J. Box.  Some Experiences with a Nonlinear Experimental Design Criterion , 1970 .

[17]  David R. Cox.  The analysis of binary data , 1970 .

[18]  P. McCullagh,et al.  Generalized Linear Models , 1972, Predictive Analytics.

[19]  W. J. Studden,et al.  Theory Of Optimal Experiments , 1972 .

[20]  H. Wynn.  Results in the Theory and Construction of D‐Optimum Experimental Designs , 1972 .

[21]  D. G. Watts,et al.  BAYESIAN ESTIMATION AND DESIGN OF EXPERIMENTS FOR GROWTH RATES WHEN SAMPLING FROM THE POISSON DISTRIBUTION , 1972 .

[22]  D. Burkholder.  Distribution Function Inequalities for Martingales , 1973 .

[23]  W. G. Cochran.  Experiments for Nonlinear Functions , 1973 .

[24]  H. Chernoff.  Approaches in Sequential Design of Experiments , 1973 .

[25]  W. G. Cochran.  Experiments for Nonlinear Functions (R.A. Fisher Memorial Lecture) , 1973 .

[26]  W. J. Hill,et al.  Design of Experiments for Subsets of Parameters , 1974 .

[27]  E. Läuter.  Experimental design in a class of models , 1974 .

[28]  G. Hohmann,et al.  On sequential and nonsequential D‐optimal experimental design , 1975 .

[29]  I. Ford.  Optimal static and sequential design : a critical review , 1976 .

[30]  H. Wynn,et al.  The Convergence of General Step-Length Algorithms for Regular Optimum Design Criteria , 1978 .

[31]  V. Borkar,et al.  Adaptive control of Markov chains, I: Finite parameter set , 1979 .

[32]  V. Borkar,et al.  Adaptive control of Markov chains , 1979 .

[33]  Peter D. H. Hill.  D-Optimal Designs for Partially Nonlinear Regression Models , 1980 .

[34]  S. Silvey,et al.  A sequentially constructed design for estimating a nonlinear parametric function , 1980 .

[35]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[36]  D. G. Watts,et al.  Relative Curvature Measures of Nonlinearity , 1980 .

[37]  T. Sweeting.  Uniform Asymptotic Normality of the Maximum Likelihood Estimator , 1980 .

[38]  Y. M. El-Fattah.  Gradient approach for recursive estimation and control in finite Markov chains , 1981, Advances in Applied Probability.

[39]  D. G. Watts,et al.  Parameter Transformations for Improved Approximate Confidence Regions in Nonlinear Least Squares , 1981 .

[40]  Y. M. El-Fattah,et al.  Recursive Algorithms for Adaptive Control of Finite Markov Chains , 1981 .

[41]  Changbao Wu,et al.  Asymptotic Theory of Nonlinear Least Squares Estimation , 1981 .

[42]  A. Atkinson.  Developments in the Design of Experiments, Correspondent Paper , 1982 .

[43]  A. C. Atkinson,et al.  Developments in the Design of Experiments , 1982 .

[44]  T. Lai,et al.  Least Squares Estimates in Stochastic Regression Models with Applications to Identification and Control of Dynamic Systems , 1982 .

[45]  P. Kumar,et al.  Optimal adaptive controllers for unknown Markov chains , 1982 .

[46]  P. Kumar,et al.  A new family of optimal adaptive controllers for Markov chains , 1982 .

[47]  V. Borkar,et al.  Identification and adaptive control of Markov chains , 1982 .

[48]  D. G. Watts,et al.  Accounting for Intrinsic Nonlinearity in Nonlinear Regression Parameter Inference Regions , 1982 .

[49]  Mitsuo Sato,et al.  Learning control of finite Markov chains with unknown transition probabilities , 1982 .

[50]  Khidir M. Abdelbasit,et al.  Experimental Design for Binary Data , 1983 .

[51]  D. Bates.  The Derivative of |X′X| and Its Uses , 1983 .

[52]  T. Sweeting.  On estimator efficiency in stochastic processes , 1983 .

[53]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[54]  D. Bates.  The Derivative of IX'XI and Its Uses , 1983 .

[55]  A. Khuri.  A Note on D-Optimal Designs for Partially Nonlinear Regression Models , 1984 .

[56]  C. Z. Wei.  Asymptotic Properties of Least-Squares Estimates in Stochastic Regression Models , 1985 .

[57]  T. Caliński,et al.  Linear Statistical Inference , 1985 .

[58]  Changbao Wu,et al.  Asymptotic inference from sequential design in a nonlinear situation , 1985 .

[59]  D. G. Watts,et al.  A Quadratic Design Criterion for Precise Estimation in Nonlinear Regression Models , 1985 .

[60]  Patchigolla Kiran Kumar,et al.  A Survey of Some Results in Stochastic Adaptive Control , 1985 .

[61]  D. Titterington,et al.  Inference and sequential design , 1985 .

[62]  Optimal design for an inverse Gaussian regression model , 1986 .

[63]  S. Minkin.  Optimal Designs for Binary Data , 1987 .

[64]  A. Gallant,et al.  Nonlinear Statistical Models , 1988 .

[65]  B. L. S. Prakasa Rao.  Asymptotic theory of statistical inference , 1987 .

[66]  Douglas M. Bates,et al.  Nonlinear Regression Analysis and Its Applications , 1988 .

[67]  M. K. Khan,et al.  On d-optimal designs for binary data , 1988 .

[68]  W. McCormick,et al.  Strong consistency of the MLE for sequential design problems , 1988 .

[69]  S. Schwartz,et al.  An accelerated sequential algorithm for producing D -optimal designs , 1989 .

[70]  Michael Woodroofe,et al.  Very weak expansions for sequentially designed experiments: linear models , 1989 .

[71]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[72]  André I. Khuri,et al.  Response surface methodology: 1966–1988 , 1989 .

[73]  K. Chaloner.  Bayesian design for estimating the turning point of a quadratic regression , 1989 .

[74]  A. Pázman.  On information matrices in nonlinear experimental design , 1989 .

[75]  K. Chaloner,et al.  Optimal Bayesian design applied to logistic regression experiments , 1989 .

[76]  D. M. Titterington,et al.  Recent advances in nonlinear experiment design , 1989 .

[77]  Christos P. Kitsos,et al.  Fully sequential procedures in nonlinear design problems , 1989 .

[78]  Dieter Rasch,et al.  Optimum experimental design in nonlinear regression , 1990 .

[79]  Anthony C. Atkinson,et al.  Optimum Experimental Designs , 1992 .