Varying-Coefficient Models Checking With Berkson Measurement Errors

Varying-coefficient models (VCM) are a useful extension of classical linear models. It was propsed by Cleveland, Grosse and Shyu, then discussed by Hastie and Tibshirani in detail. Up to now, it has made great influence in our world. In theory, the researchs of VCM has been given most atten-tion recently; Furthermore,extensive applications of VCM into biometrics and medicine has implemented successfully. But in fact, the feasibility of VCM is poor. Many researchers deal with it according to circumstances.Usually, in practice the predictor vector U is assumed to be observable. But in many experiments, it is expensive or impossible to observe U.Instead, a proxy or a manifest. Z of U can be measured.In these cases, if the amount Z is properly calibrated, then the actual predictor vector U will vary around Z randomly, so that in average the random variation U-Z will be zcro.In the situation mentioned above, a reasonable model for the measurement errors is the so-called Berkson model(see Fuller): U=Z+ηwhereηis the unobserved random measurement error which is assumed to be independent of the observed predictor variable Z.Considering there extist some errors we cannot ignore it and thus we propose a new model:Variable Coefficient Errors with Berkson Measurement Errors.In this paper, we consider a varying-coefficient model with Berkson mea-surement errors which has following form: where X=(X1,…, Xp)T,α(U)= (α1(U),…,αp(U))T,εandηare random errors with Eε=0. Eη=0, Var(ε)=σ2,α(·) are unknown functions with same smoothness, Z is the observable d-dimensional control variable, All three variablesε,ηand Z are assumed to be mutually independent, the variables X and Z are independent too。For this model,the problems of interest here are to test the two hypothesis:1. H0*:α(·)=αθ(·), whereαθ(·) is a specified function with the unknow parameterθ∈(?) Rp, H1*:H0 is not true;2.H0:α(·) =α, the amountαis a constant. H1:H0 is not true.For the two problems,we apply the generalized likelihood ratio and the empirical likelihood to test for the new testing problem under the deformated model, we demonstrate that a class of the generalized likelihood ratio statis-tics based on some appropriate nonparametric estimators follow asymptoticallyχ2-distributions under null hypotheses,and the empirical likelihood method has the advantage that the asymptotic null hypotheses is chi-squared.we present the confidence regions or confidence interval from the asymptotical result...