International Journal of Nonlinear Analysis and Applications

Nonparametric Bayesian optimal designs for unit exponential regression model with respect to prior processes (with Polya Urn scheme as the base measure)

Document Type : Research Paper

Authors

Department of Statistics, Razi University, Kermanshah, Iran

Abstract

Nonlinear regression models find extensive applications across various scientific disciplines. It is crucial to accurately fit the optimal nonlinear model while taking into account the biases inherent in the Bayesian optimal design. By utilizing the Dirichlet process as a prior, we present a Bayesian optimal design. The Dirichlet process serves as a fundamental tool in the exploration of Nonparametric Bayesian inference, offering multiple representations that are well-suited for application. This research paper introduces a novel one-parameter model, referred to as the "Unit-Exponential distribution", specifically designed for the unit interval. Additionally, we employ a representation to approximate the D-optimality criterion, considering the Dirichlet process as a functional tool. Through this approach, we aim to identify a Nonparametric Bayesian optimal design.

Keywords

[1] M. Aminnejad and H. Jafari, Bayesian A and D-optimal designs for gamma regression model with inverse link function, Commun. Statist. Simul. Comput. 46 (2017), 8166–8189.
[2] A.C. Atkinson, A.N. Donev, and R.D. Tobias, Optimum Experimental Design, With SAS, Oxford University Press, Oxford, 2007.
[3] D. Blackwell, Discreteness of Ferguson selections, Ann. Statist. 1 (1973), 356–358.
[4] L. Bondesson, On simulation from infinitely divisible distributions, Adv. Appl. Probab. 14 (1982), no. 4, 855–869.
[5] I. Burghaus and H. Dette, Optimal designs for nonlinear regression models with respect to non-informative priors, J. Statist. Plann. Infer. 154 (2014), 12–25.
[6] K.M. Chaloner and G.T. Duncan, Assessment of a beta prior distribution: PM elicitation, J. Royal Statist. Soc.: Ser. D (The Statistician) 32 (1983), no. 1-2, 174–180.
[7] K. Chaloner and K. Larntz, Optimal Bayesian design applied to logistic regression experiments, J. Statist. Plann. Infer. 21 (1989), 191–208.
[8] K. Chaloner and I. Verdinelli, Bayesian experimental design: A review, Statist. Sci. 10 (1995), 273–304.
[9] H. Chernoff, Locally optimal designs for estimating parameters, Ann. Math. Statist. 24 (1953), no. 4, 586–602.
[10] H. Dette and H.M. Neugebauer, Bayesian D-optimal designs for exponential regression models, J. Statist. Plann. Infer. 60 (1997), no. 2, 331–349.
[11] V.V. Fedorov and S.L. Leonov, Optimal Design for Nonlinear Response Models, CRC Press, 2013.
[12] T.S. Ferguson, A Bayesian analysis of some nonparametric problems, Ann. Statist. 1 (1973), no. 1, 209–230.
[13] D. Firth and J. Hinde, On Bayesian D-optimum design criteria and the equivalence theorem in nonlinear models, J. Royal Statist. Soc. B 59 (1997), no. 4, 793–797.
[14] S. Mukhopadhyay and L.M. Haines, Bayesian D-optimal designs for the exponential growth model, J. Statist. Plann. Infer. 44 (1995), no. 3, 385–397.
[15] P. Parsamaram and H. Jafari, Bayesian D-optimal Design for the logistic regression model with exponential distribution for random intercept, J. Statist. Comput. Simul. 86 (2016), no. 10, 1856–1868.
[16] L. Pronzato and E. Walter, Robust experiment design via stochastic approximation, Math. Biosci. 75 (1985), no. 1, 103–120.
[17] J. Sethuraman, A constructive definition of Dirichlet priors, Statist. Sinica 4 (1994), 639–650.
[18] M. Zarepour and L. Al Labadi, On a rapid simulation of the Dirichlet process, Statist. Probab. Lett. 82 (2012), no. 5, 916–924.
International Journal of Nonlinear Analysis and Applications
Volume 17, Issue 5
May 2026
Pages 51-59
  • Receive Date: 22 June 2024
  • Accept Date: 18 September 2024