Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Statistics and Actuarial Sciences

Supervisor

Dr. Ricardas Zitikis

Abstract

This thesis focuses on discussing non-parametric estimators and their asymptotic behaviors for indices developed to characterize bi-variate time series. There are typically two types of indices depending on whether the distributional information is involved. For the indices containing the distributional information of the bivariate stationary time series, we particularly focus on the index called the tail order of maximal dependence (TOMD), which is an improvement of the tail order. For the indices without distributional information of the bivariate time series, we focus on an anomaly detection index for univariate input-output systems.

This thesis integrates three articles. The first article (Chapter 2) proposes the \emph{average block-minima estimator} for the TOMD and discusses theoretical aspects of this estimator under the independently identically distribution (i.i.d.) assumption, including asymptotic behavior and bias reduction. The performance of this estimator is justified by simulation studies using Marshall-Olkin copula and generalized Clayton copula, respectively. The second article (Chapter 3) examines the performance of the \emph{average block-minima estimator} on stationary bi-variate time series using simulation studies. Applications of the estimator on three groups of financial assets are employed to illustrate how the estimation method could be used in practice. The third article (Chapter 4) generalizes an existing anomaly detection index for input-output systems with i.i.d. inputs to those with stationary inputs. Theoretical evidence and illustrative examples are provided to validate the performance of the existing index for systems with stationary inputs.

Summary for Lay Audience

Bi-variate time series is ubiquitous in real life such as financial risk management and reliability assessment of input-output systems. Depending on the issue of interests, various indices are proposed for bi-variate time series to serve the purpose of evaluation or comparison. Among all these indices, we are particularly interested in the following two: 1) the tail order of maximal dependence (TOMD); 2) an index of anomaly detection. The TOMD is an improvement of the existing tail order which is used to measure the extreme co-movements of random variables such as investment returns of assets. The index of anomaly detection is used to examine if the inputs and outputs of systems with certain original order contain persistent anomalies that may not change the regime such as network intrusion. In this research, we develop non-parametric estimation method for the TOMD and index of anomaly detection when the underlying data-generating process is bi-variate time series.

This research consists of three articles. The first article proposes the average block-minima estimator of the TOMD for independently identically distributed random pairs. The performance of the estimator is justified by both theoretical analysis and simulation studies. The second article examines the performance of the average block-minima estimator of the TOMD for stationary bi-variate time series using simulation studies. The third article extends the existing results for the anomaly detection index to the systems with bi-variate stationary inputs with theoretical validation.

Download

Share

COinS