Date of Award

Fall 2023

Project Type

Dissertation

Program or Major

Statistics

Degree Name

Doctor of Philosophy

First Advisor

Linyuan Li

Second Advisor

Deniz Ozabaci

Third Advisor

Qi Zhang

Abstract

The world of time series analysis is predominantly focused on nonstationary time series, which exhibit various patterns and behaviors that researchers across different industries aim to analyze. Decomposing nonstationary time series into a trend component and a stationary process is a common approach, with trend and stationary process estimation achieved through parametric and nonparametric methods.

Polynomial regression and nonparametric methods, such as Kernels and B-Splines, are employed to estimate the trend. Once the trend is estimated, the residuals are obtained by subtracting the estimated trend from the data, allowing for the fitting of an ARMA stationary process. However, the estimation of the trend using these methods can be unstable since it depends on the unknown underlying stationary process.

To address these challenges, Kauermann et al. (2011) introduce a robust approach using P-splines within linear mixed models. This approach provides an estimated trend and an ARMA(p, q) stationary process, enabling inference in the analysis. Building upon this work, Shao and Yang (2017) propose a modified 2-step method using B-splines to estimate the trend and autocorrelated errors separately. This method involves removing the trend first and then estimating the errors independently.

In this thesis, we investigate the linear mixed model and 2-step methods of inference on nonstationary time series and aim to propose a new approach for simultaneous and robust estimation of the trend and stationary process. Our approach offers several advantages that are currently absent in the linear mixed model and 2-step methods. It provides stable estimation of the trend, the white noise variance, and allows for the estimation of higher order autocorrelated ARMA(p, q) errors, which are currently computationally limited in the linear mixed model method.

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