Nonparametric change-point analysis, invariance principles for multivariate Student processes, and asymptotic theory in linear errors-in-variables models

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This Proposal consists of parts 1)-3) that deal respectively with three interrelated areas of statistics and probability theory that are listed in the title. It is based on recent advances made in these areas in the applicant's research publications.******Change-point problems arise in a large variety of sciences when detecting if a sequence of chronologically ordered data could be viewed as homogeneous in a stochastic sense. Testing for changes in the (theoretical) mean (average) of a data sequence is one basic situation to which other change-point problems can be reduced. Part 1) a) of Proposal is to deal with nonparametric tests for at most one such change when the data are stochastically independent and follow general enough conditions. This research will provide guidance to practitioners on the performance and use of the available tests for at most one change in the mean and will extend the type of such tests recently proposed by the applicant for multivariate and other kinds of data.******In linear errors-in-variables models (EIVM) two variables are linearly related and are observed with measurement errors that complicate statistical inference in these models. Part 1) b) of Proposal is to develop procedures for nonparametric detection of a possible change in the slope and intercept in linear EIVM's while collecting data from these models. The proposed change-point problems occur naturally in data analysis of numerous applications of EIVM's, which include virtually all research areas, but have not yet been studied.******Part 2) a) of Proposal intends to obtain the full, or at least a rather general partial, resolution of a conjecture on a characterization of the asymptotic normality property of the multivariate Student t-statistic. The Student t-statistic has played a central role in inferential statistics. Studying when it is asymptotically standard normal is important for various applications such as constructing approximate confidence intervals for a population mean from large samples of data.******The anticipated invariance principles for the multivariate Student (stochastic) process (a generalization of the multivariate Student t-statistic) of part 2) b) of Proposal will contribute to recent advances in limit theorems for self-normalized processes of probability theory and can become a key tool for developing nonparametric tests for a possible change in the mean in a sequence of some multivariate data.******For linear EIVM's under some new conditions, part 3) of Proposal is to obtain asymptotic results for inference from large samples of data with possibly infinite variances from these models. The results will be based on the least squares estimators from linear regression and invariance principles for self-normalized processes of independent random variables. They will lead to first time, easily available large-sample approximate confidence regions/intervals for the slope and intercept in such EIVM's.