This dissertation contains three essays. They are related to the exponential series estimation of copulas and the application of parametric copulas in financial econometrics. Chapter II proposes a multivariate exponential series estimator (ESE) to estimate copula density nonparametrically. The ESE attains the optimal rate of convergence for nonparametric density. More importantly, it overcomes the boundary bias of copula estimation. Extensive Monte Carlo studies show the proposed estimator outperforms kernel and log-spline estimators in copula estimation. Discussion is provided regarding application of the ESE copula to Asian stock returns during the Asian financial crisis. The ESE copula complements the existing nonparametric copula studies by providing an alternative dedicated to the tail dependence measure. Chapter III proposes a likelihood ratio statistic using a nonparametric exponential series approach. The order of the series is selected by Bayesian Information Criterion (BIC). I propose three further modifications on my test statistic: 1) instead of putting equal weight on the individual term of the exponential series, I consider geometric and exponential BIC average weights; 2) rather than using a nested sequence, I consider all subsets to select the optimal terms in the exponential series; 3) I estimate the likelihood ratio statistic using the likelihood cross-validation. The extensive Monte Carlo simulations show that the proposed tests enjoy good finite sample performances compared to the traditional methods such as the Anderson-Darling test. In addition, this data-driven method improves upon Neyman’s score test. I conclude that the exponential series likelihood ratio test can complement the Neyman’s score test. Chapter IV models and forecasts S&P500 index returns using the Copula-VAR approach. I compare the forecast performance of the Copula-VAR model with a classical VAR model and a univariate time series model. I use this approach to forecast S&P500 index returns. I apply a modified Diebold-Mariano test to test the equality of mean squared forecast errors and utilize a forecast encompassing test to evaluate forecasts. The findings suggest that allowing a more flexible specification in the error terms using copula tends improve the forecast accuracy. I also demonstrate combined forecasts improved forecasts accuracy over individual models.

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.

This thesis studies the nonlinear characteristics of microwave devices by Volterra series because it can analyze the nonlinear devices with memory. And a nonlinear model was established by measurement data for Volterra series analysis. This content is composed of three parts. The first part devote to introduce the nonlinear phenomenon and theories of nonlinear analysis. The difference between power series and Volterra series could be realized by deriving them. The second part is to introduce the physical characteristics of pHEMTs and demonstrate the procedure of establishing small signal model and fitting nonlinear equations of currents and capacitances, and a process of nonlinear model analysis by Volterra series is shown. The third part is to describe the experimental arrangements and analyze nonlinear characteristics of pHEMTs actually with above methods. And the relationship among nonlinear sources was discussed. The device was fabricated by WIN 0.15£gm InGaAs process and measured by on wafer measurements.

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

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

Thesis (doctoral)--Georg-August-Universität zu Göttingen, 1920. / Vita. Includes bibliographical references (p. [53]-54).

Thesis (doctoral)--Faculté des sciences de Lund, 1918. / Includes bibliographical references.

Thesis (doctoral)--Friedrich-Wilhelms Universität zu Berlin, 1878. / Vita.

This thesis aims at investigating different forms of residuals from a general time series model with conditional mean and conditional variance fitted by the Gaussian quasi-maximum likelihood method. We investigated the limiting distributions of autocorrelation and partial autocorrelation functions under different forms of residuals. Based on them we devised some individual portmanteau tests and two mixed portmanteau tests. We started by exploring the asymptotic normalities of the residual autocorrelation functions, the squared residual autocorrelation functions and absolute residual autocorrelation functions from the fitted time series model. This leads to three individual portmanteau tests. Then we generalized them to their counterparts of partial autocorrelation functions, and this results in another three individual portmanteau tests. We carried out simulations studies to compare the six individual portmanteau tests and find that some tests are sensitive to mis-specification in the conditional mean while other tests are effective to detect mis-specification in the conditional variance. However, for the case that the mis-specifications happen in both conditional mean and variance, none of these individual portmanteau tests present remarkable power. Based on this, we continued to investigate the joint limiting distributions of the residual autocorrelation functions and absolute residual autocorrelation functions of the fitted model since the former one is powerful to identify mis-specification in the conditional mean and the latter one works well when the conditional variance is mis-specified. This leads to two mixed portmanteau tests for diagnostic checking of the fitted model. Simulation studies are carried out to check the asymptotic theory and to assess the performance of the mixed portmanteau tests. It shows that the mixed portmanteau tests have the power to detect mis-specification in the conditional mean and conditional variance respectively while it is feasible to detect both of them. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

The subject of time series analysis has drawn significant attentions in recent years, since it is of tremendous interest to practitioners, as well as to academic researchers, to make statistical inferences and forecasts of future values of the interested variables. To do forecasting, parametric models are often required to describe the patterns of the observed data set. In order to describe data adequately, such statistical models should be established based on fundamental principles. Two threshold-type time series models, the buffered threshold autoregressive (BAR) model and the threshold moving-average (TMA) model are studied in this thesis. The most important contribution of this thesis is the extension of the classical threshold models via regime switching mechanisms that exhibit hysteresis to a new model called the buffered threshold model. For this type of new models, there is a buffer zone for the regime switching mechanism. The self-exciting buffered threshold autoregressive model has been thoroughly studied: a sufficient condition is given for the geometric ergodicity of the two-regime BAR process; the conditional least squares estimation is considered for the parameter estimation of the BAR model, and asymptotic properties including strong consistency and asymptotic distributions of the least square estimators are also derived. Monte Carlo experiments are conducted to give further support to the methodology developed for the new model. Two empirical examples are used to demonstrate the importance of the BAR model. Potential extensions for the basic buffer processes are discussed as well. Such extensions are expected to follow the development of classical threshold model and are motivated by their relationships with phenomena in the physical sciences. The proposed buffer process is more general than the classical threshold model, and it should be able to capture more nonlinear features exhibited by this nonlinear world than its predecessor. Although the theoretical understanding of the model is still at its infancy, it is believed that the buffer process will provide both researchers and practitioners with a useful tool to understand the nonlinear world. Moreover, some statistical properties of the threshold moving-average models are studied. Computer simulations have been extensively used, and some mathematical interpretation is attempted in the light of some existing research works. The model-building procedure for the TMA models is also reviewed. The effectiveness of some classical information criteria in selecting the correct TMA model is studied. A goodness-of-fit test is derived which would be useful in diagnostic checking the fitted TMA models. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy