Electronic Theses and Dissertations

Date of Award

2017

Document Type

Dissertation

Degree Name

Ph.D. in Mathematics

Department

Mathematics

First Advisor

Hailin Sang

Second Advisor

Hailin Sang

Third Advisor

Walt Mayer

Abstract

A large class of time series processes can be modeled by linear processes, including a subset of the fractional ARIMA process. Transformation of linear processes is one of the most popular topics in univariate time-series analysis in recent years. In this dissertation, we study the memory properties of transformations of linear processes. Our results show that the transformations of short-memory time series still have short-memory and the transformation of long-memory time series may have different weaker memory parameters which depend on the power rank of the transformation. In particular, we provide the memory parameters of the FARIMA (p,d,q) processes. As an example, the memory properties of call option processes at different strike prices are discussed in details. When we develop the memory properties of transformation of linear processes, we use the Pearson correlation to measure the memory. Correlation is another big topic in statistics, which is used to measure the dependence of stochastic processes or random variables. Standard Gini correlation is one of the correlations to measure the dependence between random variables with heavy tailed distributions. However, the asymmetry of Gini covariance and correlation brings a substantial difficulty in interpretation. In this dissertation, we propose a symmetric Gini-type covariance and correlation (ρg) based on the joint rank function. The proposed correlation ρg is symmetric and is more robust than the Pearson correlation but less robust than the Kendall's τ correlation in terms of influence functions. Furthermore, we establish the relationship between ρg and the linear correlation ρ for a class of random vectors in the family of elliptical distributions, which allows us to estimate ρ based on estimation of ρg. We compare asymptotic efficiencies of linear correlation estimators based on the symmetric Gini, and the proposed measure ρg shows superior finite sample performance, which makes it attractive in applications.

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