Date of Award
Doctor of Philosophy (PhD)
School of Mathematical and Statistical Sciences
Dr. Colin Gallagher
Dr. Andrew Brown
Dr. Christopher McMahan
Dr. Qiong Zhang
This dissertation explored the idea of penalized method in estimating the autocorrelation (ACF) and partial autocorrelation (PACF) in order to solve the problem that the sample (partial) autocorrelation underestimates the magnitude of (partial) autocorrelation in stationary time series. Although finite sample bias corrections can be found under specific assumed models, no general formulae are available. We introduce a novel penalized M-estimator for (partial) autocorrelation, with the penalty pushing the estimator toward a target selected from the data. This both encapsulates and differs from previous attempts at penalized estimation for autocorrelation, which shrink the estimator toward the target value of zero. Unlike the regression case, in which the least squares estimator is unbiased and shrinkage is used to reduce mean squared error by introducing bias, in the autocorrelation case the usual estimator has bias toward zero. The penalty can be chosen so that the resulting estimator of autocorrelation is asymptotically normally distributed. Simulation evidence indicates that the proposed estimators of (partial) autocorrelation tend to alleviate the bias and reduce mean squared error compared with the traditional sample ACF/PACF, especially when the time series has strong correlation.
One application of the penalized (partial) autocorrelation estimator is portmanteau tests in time series. Target and tuning parameters can be selected to improve time series Portmanteau tests--shrinking small magnitude correlations toward zero controls type I error, while increasing larger magnitude correlations improves power. Specific data based choices for target and tuning parameters are provided for general classes of time series goodness of fit tests. Asymptotic properties of the proposed test statistics are obtained. Simulations show power is improved for all of the most prevalent tests from the literature and the proposed methods are applied to data.
Another application of the penalized ACF/PACF considered in this dissertation is the optimal linear prediction of time series. We exploit ideas from high-dimensional autocorrelation matrix estimation and use tapering and banding, as well as a regularized Durbin-Levinson algorithm to derive new predictors that are based on the penalized correlation estimators. We show that the proposed estimators reduce the error in linear prediction of times series. The performance of the proposed methods are demonstrated on simulated data and applied to data.
Tan, Xiyan, "Penalized Estimation of Autocorrelation" (2022). All Dissertations. 3016.