published_or_final_version / abstract / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

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The objective of this dissertation is to develop a suitable statistical methodology for parameter estimation in long memory process. Time series data with complex covariance structure are shown up in various fields such as finance, computer network, and econometrics. Many researchers suggested the various methodologies defined in different domains: frequency domain and time domain. However, many traditional statistical methods are not working well in complicated case, for example, nonstationary process. The development of the robust methodologies against nonstationarity is the main focus of my dissertation. We suggest a wavelet-based Bayesian method which shares good properties coming from both wavelet-based method and Bayesian approach. To check the robustness of the method, we consider ARFIMA(0, d, 0) with linear trend. Also, we compare the result of the method with that of several existing methods, which are defined in different domains, i.e. time domain estimators, frequency domain estimators. Also, we apply the method to functional magnetic resonance imaging (fMRI) data to find some connection between brain activity and long memory parameter. Another objective of this dissertation is to develop a wavelet-based denoising technique when there is heterogeneous variance noise in high throughput data, especially protein mass spectrometry data. Since denoising technique pretty much depends on threshold value, it is very important to get a proper threshold value which involves estimate of standard deviation. To this end, we detect variance change point first and get suitable threshold values in each segment. After that, we apply local wavelet thresholding to each segment, respectively. For comparison, we consider several existing global thresholding methods.

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 84-99).

With the increasingly economic activities, people have more and more interest in econometric models. There are two mainstream econometric models which are very popular in recent decades. One is quantile autoregressive (QAR) model which allows varying-coefficients in linear time series and greatly promotes the ranges of regression research. The first topic of this thesis is to focus on the modeling of QAR model. We propose two important measures, quantile correlation (QCOR) and quantile partial correlation (QPCOR). We then apply them to QAR models, and introduce two valuable quantities, the quantile autocorrelation function (QACF) and the quantile partial autocorrelation function (QPACF). This allows us to extend the Box-Jenkins three-stage procedure (model identification, model parameter estimation, and model diagnostic checking) from classical autoregressive models to quantile autoregressive models. Specifically, the QPACF of an observed time series can be employed to identify the autoregressive order, while the QACF of residuals obtained from the model can be used to assess the model adequacy. We not only demonstrate the asymptotic properties of QCOR, QPCOR, QACF and PQACF, but also show the large sample results of the QAR estimates and the quantile version of the Ljung- Box test. Moreover, we obtain the bootstrap approximations to the distributions of parameter estimators and proposed measures. Simulation studies indicate that the proposed methods perform well in finite samples, and an empirical example is presented to illustrate the usefulness of QAR model. The other important econometric model is autoregressive conditional duration (ACD) model which is developed with the purpose of depicting ultra high frequency (UHF) financial time series data. The second topic of this thesis is designed to incorporate ACD model with one of the extreme value distributions, i.e. Fréchet distribution. We apply the maximum likelihood estimation (MLE) to Fréchet ACD models and derive its generalized residuals for model adequacy checking. It is noteworthy that simulations show a relative greater sensitiveness in the linear parameters to sampling errors. This phenomenon successfully reflects the skewness of the Fréchet distribution and suggests a method to practitioners in proceeding model accuracy. Furthermore, we present the empirical sizes and powers for Box-Pierce, Ljung-Box and modified Box-Pierce statistics as comparisons of the proposed portmanteau statistic. In addition to the Fréchet ACD, we also systematically analyze theWeibull ACD, where the Weibull distribution is the other nonnegative extreme value distribution. The last topic of the thesis explains the estimation and diagnostic checking the Weibull ACD model. By investigating the MLE in this model, there exhibits a slight sensitiveness in linear parameters. However, there is an obvious phenomenon on the trade-off between the skewness of Weibull distribution and the sampling error when the simulations are conducted. Moreover, the asymptotic properties are also studied for the generalized residuals and a goodness-of-fit test is employed to obtain a portmanteau statistic. Through the simulation results in size and power, it shows that Weibull ACD is superior to Fréchet ACD in specifying the wrong model. This is meaningful in practice. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

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