Abstract
Oracally efficient estimation and an asymptotically accurate simultaneous confidence band are established for the nonparametric link function in the partially linear single-index models for longitudinal data. The proposed procedure works for possibly unbalanced longitudinal data under general conditions. The link function estimator is shown to be oracally efficient in the sense that it is asymptotically equivalent in the order of $n^{-1/2}$ to that with all true values of the parameters being known oracally. Furthermore, the asymptotic distribution of the maximal deviation between the estimator and the true link function is provided, and hence a simultaneous confidence band for the link function is constructed. Finite sample simulation studies are carried out which support our asymptotic theory. The proposed SCB is applied to analyze a CD4 data set.
Highlights
There has been substantial interest in semiparametric partially linear models in the last three decades as they enjoy the advantages of both the flexibility of nonparametric modeling and easy interpretation of parametric modeling
Performance of the estimates of the parameters measured by the averaged mean squared error (MSE) for the data generated by normally distributed covariates based on 1000 replications under the correct specification of the underlying correlation structure
Performance of the estimates of the parameters measured by the averaged mean squared error (MSE) for the data generated by normally distributed covariates based on 1000 replications under misspecification of the underlying correlation structure
Summary
There has been substantial interest in semiparametric partially linear models in the last three decades as they enjoy the advantages of both the flexibility of nonparametric modeling and easy interpretation of parametric modeling. Process, X (t) and Z (t) are p and q dimensional covariates, and ε (t) is a random error process For this model in the context of longitudinal data, many different approaches have been studied to estimate the unknown coefficient vector Θ = (βT , θT )T and link function φ (·). The proposed SCB for the link function is innovative and useful since it is the first of its kind for possibly unbalanced longitudinal data in the partially linear single-index modeling structure with sound theoretical justifications. It provides satisfactory numerical performance in finite sample sizes.
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