Bayesian D-optimal designs on a fixed number of design points for heteroscedastic polynomial models

SUMMARY We consider design issues in a polynomial regression model where the variance of the response depends on the independent variable exponentially. However, this dependence is not known precisely and additional parameters are required in the model. Our design criteria permit various subsets of the parameters to be estimated with different emphasis. Bayesian D-optimal designs on a compact interval, with the number of support points restricted to be one more than the degree of the polynomial, are found analytically for a large class of priors. These designs may or may not be optimal within the class of all designs, depending on the prior distribution.

[1]  Anthony C. Atkinson,et al.  D-Optimum Designs for Heteroscedastic Linear Models , 1995 .

[2]  W. J. Studden,et al.  OPTIMAL DESIGNS FOR WEIGHTED POLYNOMIAL REGRESSION USING CANONICAL MOMENTS , 1982 .

[3]  Carl Lee,et al.  Constrained optimal designs for regressiom models , 1987 .

[4]  W. J. Studden,et al.  Optimal Designs for Trigonometric and Polynomial Regression Using Canonical Moments , 1985 .

[5]  W. J. Studden.  $D_s$-Optimal Designs for Polynomial Regression Using Continued Fractions , 1980 .

[6]  Holger Dette,et al.  OPTIMAL BAYESIAN DESIGNS FOR MODELS WITH PARTIALLY SPECIFIED HETEROSCEDASTIC STRUCTURE , 1996 .

[7]  W. J. Studden,et al.  Tchebycheff Systems: With Applications in Analysis and Statistics. , 1967 .

[8]  M. Skibinsky.  Principal representations and canonical moment sequences for distributions on an interval , 1986 .

[9]  W. J. Studden.  Some Robust-Type D-Optimal Designs in Polynomial Regression , 1982 .

[10]  F. Pukelsheim.  Optimal Design of Experiments , 1993 .

[11]  Holger Dette,et al.  The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis , 1997 .

[12]  Holger Dette,et al.  Bayesian D-optimal designs for exponential regression models , 1997 .

[13]  K. Chaloner,et al.  The Equivalence of Constrained and Weighted Designs in Multiple Objective Design Problems , 1996 .

[14]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[15]  Stephen Wolfram,et al.  Mathematica: a system for doing mathematics by computer (2nd ed.) , 1991 .

[16]  W. J. Studden,et al.  On an extremal problem of Feje´r , 1988 .

[17]  Ingram Olkin,et al.  Jack Carl Kiefer Collected Papers III: Design of Experiments. , 1987 .

[18]  Weng Kee Wong,et al.  On the Equivalence of Constrained and Compound Optimal Designs , 1994 .

[19]  K. Chaloner,et al.  Bayesian Experimental Design: A Review , 1995 .

[20]  Kathryn Chaloner,et al.  A note on optimal Bayesian design for nonlinear problems , 1993 .

[21]  Carl Lee,et al.  Constrained optimal designs , 1988 .

[22]  Holger Dette,et al.  Bayesian optimal one point designs for one parameter nonlinear models , 1996 .

[23]  R. Fateman,et al.  A System for Doing Mathematics by Computer. , 1992 .

[24]  S. Weisberg,et al.  Diagnostics for heteroscedasticity in regression , 1983 .