This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The principal investigator (P.I.) will study three topics in measurement error models, develop relevant methodologies and analyze their corresponding properties and performances.
The first topic concerns functional models in the situation when the main model contains unspecified error, hence is a semiparametric model by itself. The P.I. will study the structure and interaction of the two nonparametric components, the unknown error distribution and the unknown latent variable distribution, and propose an operation to best treat each of them. The general approach is geometry based. The resulting estimating procedure possesses robustness to model misspecification in both components, and allows to achieve optimal estimation efficiency. The asymptotic consistency and normality will be demonstrated both theoretically and in numerical examples. The second topic concerns the model goodness-of-fit test in measurement error models. The P.I. will propose a pseudo-score type methodology. She will demonstrate that the new testing procedure is feasible in accommodating the computational issues specific in such models, and has the desired consistency and power property. In addition, the optimal power property associated with the usual profiling estimation procedure can be equivalently achieved via projection. She will also study the relation between the Wald test and the pseudo-score test and demonstrate their equivalence in a wider range than previously known in literature. The third topic concerns the small sample performance in measurement error models. Existing literature has indicated that the first order asymptotics in measurement error models often require very large sample size to show its relevancy. The P.I. will tackle this problem using a saddle point approximation technique, hence achieving a higher order approximation than the classical first order theory. Because the functional measurement error model is semiparametric, yet existing saddle point approximation theory is developed and heavily relies on parametric model assumption, the P.I. will develop and study new methodology in this area.
The series of projects in this proposal will resolve some of the most fundamental issues in their most general form in measurement error models. Since errors in measurements widely present in almost all scientific fields, including health and medicine, environment and atmospheric science, finance and economics, material and chemical sciences, the new methodologies will generate wide interest and have important application in these fields. They will also provoke further studies and development in related semiparametric problems and computing methods in statistical sciences itself.
