Bin Yu Statistics with Dependent Data

1. STATISTICS WITH DEPENDENT DATA

1.1 Empirical Processes

By the mid-eighties (more than a decade after the seminal work of Vapnik and Cervonenkis, 1971), empirical processes indexed by V-C classes from iid observations were well understood and the V-C theory for the iid case was rather complete. The success of extensions of V-C theory to other stationary observations relies on carrying out the symmetrization which makes the conditioning argument work in the iid case. As part of my thesis

B. Yu, ``Some Results on Empirical Processes and Stochastic Complexity,''
Ph.D. Thesis, Dept. of Statistics, University of California at Berkeley, Apr. 1990.

I carried out successfully a symmetrization for absolutely regular stationary processes on the iid blocks of observations which are constructed to be close in distribution to the original sequence of observations. This led to a rate of convergence result and a CLT result. The rate result appeared in

B. Yu, ``Rates of convergence for empirical processes of stationary mixing sequences,''
Ann. Probab. 1994, 94--116.

The CLT was later refined and appeared with new results on U-processes in

M. Arcones and B. Yu, `` Central limit theorems for empirical and U-processes of stationary mixing sequences,'' J. Theor. Probab. 1994, 47--71.

For an extension to long-range dependent case, see

M. Arcones and B. Yu, `` Limit theorems for empirical processes under dependence,''
In Proc. in Chaos expansions, multiple Ito--Wiener integrals and their applications. 1994, 205--221.

Empirical process techniques have become by now standard tools in asymptotic analysis of statistical models. The combination of blocking and symmetrization I developed is very useful to prove rate of convergence and asymptotic normality results for absolutely regular sequences. For example, it is used in

B. Yu, `` Density estimation in the $L^\infty$ norm for dependent data with applications,''
Ann. Statist. 1993, 711--735.

for $L^\infty$-norm density estimation with absolutely regular observations such as those from the Gibbs Sampler (or Markov Chain Monte Carlo methods). Other researchers who have worked or/and are working on empirical processes from dependent data include Arcones, Massart, Doukhan, Rio, Pollard and Andrews.