Optimal Experimental Designs For Several Measurement Error Models

In the past decades,measurement error models have been widely used in the fields of biology,medicine,infectious disease,economics,finance and remote sensing to fit the data generated in these fields,so as to estimate and predict the change trend of some ran-dom phenomena.With the deepening of the application,a lot of requirements for optimal designs arise.Unlike the linear model,the experimental designs for measurement error models are often more complex.The main reasons are as follows:firstly,the existence of measurement error introduces more error variance structures into the models;Secondly,the experimental design problem for the measurement error model presents the characteristics of nonlinear model,that is,the optimal criteria are related to the estimated parameters.Thirdly,different types of measurement error models have different parameter estimation methods,so the experimental designs for measurement error models have to deal with the specific models one by one and choose different techniques.In view of this,the literature on optimal designs for measurement error models is rare,and there are still many prob-lems to be solved.In order to further enrich and improve the theory of optimal design of measurement error models,this paper will study the optimal design problems for several common measurement error models based on maximum likelihood estimation,corrected s-core function approach and minimum distance estimation.These models include functional polynomial measurement error models,functional weighted polynomial measurement error models,functional spline measurement error models with estimated knots and the general univariate Berkson measurement error models.It is well known that the estimators derived from the maximum likelihood estimation under the normal distribution generally do not satisfy the consistency when there are errors in the covariables.To overcome this problem,additional assumptions are required,usu-ally requiring that the covariance matrix of the response variable and the proxy variables(observable covariables)be known(or a multiple of the known matrix),and thus under this assumption,we study the optimal designs for functional polynomial measurement er-ror models and spline measurement error models with estimated knots.For functional polynomial measurement error models,the general equivalence theorems under the corre-sponding criteria are constructed with the help of locally and Bayesian-optimal criteria,and Chebyshev system is used to describe the characteristics of the number and value of the support points of the locally and Bayesian-optimal designs.Finally,the theoreti-cal results obtained are applied to the specific medical case,it is found that locally and Bayesian-optimal designs have significant advantages over the actual medical design in efficiency.Based on the locally-optimal criterion,the general equivalence theorem for the functional spline measurement error model with estimated knots is established.Under certain conditions,it is verified that the support points of the locally-optimal design of the model are related to the endpoints of the design region.In addition,the construction method of locally-optimal design for a special spline measurement error model is derived.If the variances of the errors are efficiency functions,the maximum likelihood estimator no longer satisfies the consistency.Then,for functional weighted polynomial measurement error models,we use the corrected score function approach to establish the correspond-ing approximate design theory on the basis of the non-concave theory,and put forward the necessary condition of the?-optimal criterion.Under two kinds of heteroscedastic structures,the ranges of the number of the support points of locally-optimal designs for the-order functional weighted polynomial measurement error models are obtained.In particular,the locally-optimal designs for the heteroscedastic linear measurement error models are uniquely determined.In addition to the functional measurement error model,another kind of measurement error model,i.e.,Berkson measurement error model,often appears in agriculture,phar-maceutical,biomedical and environmental research.Because Berkson measurement error model is quite different from the classical measurement error model in random charac-teristics,the estimation of the parameter is also completely different.According to the minimum distance estimation,a corresponding approximate design theory is established based on the non-concave theory for the general univariate Berkson measurement error model,and the necessary condition of?-optimal criterion is proposed.Under the lin-ear and exponential Berkson measurement error models,the ranges of the number of the support points of the locally-optimal designs are obtained based on the extended Cheby-shev system.The numerical simulation results show that the locally-optimal designs are much more efficient than the common equidistant designs.