On limit horizons in high dimensional inference
Soumendra N. Lahiri, Statistics Department, North Carolina State University, USA
We consider a common situation arising in many high dimensional statistical inference problems where the dimension d diverges with the sample size n and the statistic of interest is given by a function of component-wise summary statistics. The limit distribution of the statistic of interest is often influenced by an intricate interplay of underlying dependence structure of the component-wise summary statistics. Here, we introduce a new concept, called limit horizon (L.H.) that gives the boundary of the growth rate of d as a function of n where the natural approach to deriving the limit law by iterated limits works. Further, for d growing at a faster rate beyond the L.H., the natural approach breaks down. We investigate the L.H. in some specific high dimensional problems.