The focus of this chapter is on computing entropy for empirical processes. An important use of such entropy calculations is in evaluating whether a class of functions F is Glivenko-Cantelli and/or Donsker or neither. Several additional uses of entropy bounds will be discussed in Chapter 11. Some of these uses will be very helpful in Chapter 14 for establishing rates of convergence for M-estimators. An additional use of entropy bounds, one which will receive only limited discussion in this book, is in precisely assessing the asymptotic smoothness of certain empirical processes. Such smoothness results play a role in the asymptotic analysis of a number of statistical applications, including confidence bands for kernel density estimation (eg., Bickel and Rosenblatt, 1973) and certain hypothesis tests for multimodality (Polonik, 1995).
KeywordsProbability Measure Empirical Process Entropy Calculation Pointwise Limit Countable Dense Subset
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