Modern Statistical Inference for Classical Statistical Problems
- Author(s): Lei, Lihua
- Advisor(s): Bickel, Peter J
- Jordan, Michael I
- et al.
This dissertation addresses three classical statistics inference problems with novel ideas and techniques driven by modern statistics. My purpose is to highlight the fact that even the most fundamental problems in statistics are not fully understood and the unexplored parts may be handled by advances in modern statistics. Pouring new wine into old bottles may generate new perspectives and methodologies for more complicated problems. On the other hand, re-investigating classical problems help us understand the historical development of statistics and pick up the scattered pearls forgotten over the course of history.
Chapter 2 discusses my work supervised by Professor Noureddine El Karoui and Professor Peter J. Bickel on regression M-estimates in moderate dimensions. In this work, we investigate the asymptotic distributions of coordinates of regression M-estimates in the moderate $p/n$ regime, where the number of covariates $p$ grows proportionally with the sample size $n$. Under appropriate regularity conditions, we establish the coordinate-wise asymptotic normality of regression M-estimates assuming a fixed-design matrix. Our proof is based on the second-order Poincar