  We consider a joint asymptotic framework for studying semi-nonparametric
regression models where (finite-dimensional) Euclidean parameters and
(infinite-dimensional) functional parameters are both of interest. The class of
models in consideration share a partially linear structure and are estimated in
two general contexts: (i) quasi-likelihood and (ii) true likelihood. We first
show that the Euclidean estimator and (pointwise) functional estimator, which
are re-scaled at different rates, jointly converge to a zero-mean Gaussian
vector. This weak convergence result reveals a surprising joint asymptotics
phenomenon: these two estimators are asymptotically independent. A major goal
of this paper is to gain first-hand insights into the above phenomenon.
Moreover, a likelihood ratio testing is proposed for a set of joint local
hypotheses, where a new version of the Wilks phenomenon [Ann. Math. Stat. 9
(1938) 60-62; Ann. Statist. 1 (2001) 153-193] is unveiled. A novel technical
tool, called a joint Bahadur representation, is developed for studying these
joint asymptotics results.
