Inference in Increasing Dimension

Since 1960s, statisticians have started realizing the importance of characterizing the role of parameter dimension in statistical inference. For this, pioneers like Peter Huber started introducing a new feature into asymptotics: the number of parameters, p, is now allowed to increase with the sample size n. Following this track of thinking, in 1980-90's, statisticians like Stephen Portnoy and Enno Mammen developed a set of seminal results, sharply characterizing the limiting behaviors of regression estimators and exploring the growth rate boundary of p with regard to n. More recent results, aiming at tackling more general statistical problems, include He and Shao (2000) and Spokoiny (2012). In this talk, I will give a systematic review of these results, as well as disclose more new features. In particular, simple criterion will be given for guaranteeing asymptotic normality for general problems under the boundary condition p^2/n -> 0. The proof rests on Talagrand generic chaining for empirical processes. Smoothness will be shown, as always, to play the key role in analysis.