Department Seminar Series: Guang Chen, Ph.D., A Long March Towards Joint Asymptotics: My 1st Steps...

Abstract: We consider a joint asymptotic framework for studying semi-nonparametric models where (finite dimensional) Euclidean parameters and (infinite dimensional) functional parameters are both of interest. A class of generalized partially linear models is used as a prototypical example (under the penalized estimation). We first show that the Euclidean estimator and (point-wise) functional estimator, which are re-scaled at different rates, jointly converge to a Gaussian vector. This weak convergence result reveals a surprising joint asymptotics phenomenon: these two estimators become asymptotically independent while the Euclidean estimator achieves the semiparametric efficiency bound. Our first goal is to provide deep theoretical insights into the above phenomenon. A semi-nonparametric version of the Wilks phenomenon is unveiled as an interesting by-product. Our second goal is to develop more useful joint global inference for the same class of models. In particular, we find that "the inclusion of a faster convergent parametric estimator indeed affects the nonparametric global/local inference." This conclusion is against the common intuition that the parametric term (given its faster convergence rate) can be treated as if it were known. In the end, I will discuss a few open problems in this new field.