Abstract
In time series analysis, statistics based on collections of estimators computed from sub-samples play a crucial role in an increasing variety of important applications. Proving results about the joint asymptotic distribution of such statistics is challenging since it typically involves a nontrivial verification of technical conditions and tedious case-by-case asymptotic analysis. In this paper, we provide a novel technique that allows to circumvent those problems in a general setting. Our approach consists of two major steps: a probabilistic part which is mainly concerned with weak convergence of sequential empirical processes, and an analytic part providing general ways to extend this weak convergence to functionals of the sequential empirical process. Our theory provides a unified treatment of asymptotic distributions for a large class of statistics, including recently proposed self-normalized statistics and sub-sampling based p-values. In addition, we comment on the consistency of bootstrap procedures and obtain general results on compact differentiability of certain mappings that seem to be of independent interest.
Highlights
Introduction and motivationIn time series analysis, a large class of statistics can be expressed as smooth functions of estimators computed on consecutive portions of data
A large class of statistics can be expressed as smooth functions of estimators computed on consecutive portions of data
Since time series observations are naturally ordered by time, the use of such statistics has been a common theme in time series inference and examples are abundant in areas such as sequential monitoring [Chu and White (1995); Aue and Reimherr (2009)], retrospective change point detection [Csorgoand Horvath (1997); Perron (2006)] and subsampling-based inference [Politis and Romano (1994); Politis et al (1999)], among others
Summary
A large class of statistics can be expressed as smooth functions of estimators computed on consecutive portions (i.e., subsamples) of data. Returning to a more general setting, we can say that the classical delta method and a large collection of results on the behavior of general empirical processes allow to establish weak convergence results for a wide class of statistics as long as we consider a fixed, finite collection of values κ Another fundamental question that needs to be taken care of before we can apply the functional delta method is the weak convergence of the process Yn. results on weak convergence of Yn in settings where the data X1, . Section C in the appendix provides details on a test for change-points
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