In this thesis, we explore some asymptotic results in heavy-tailed theory. There are many empirical and compelling evidence in statistics that require modeling with heavy tailed observations. This thesis is divided into three parts. First, we consider a robust estimation of the mean vector for a sequence of independent and identically distributed observations in the domain of attraction of a stable law with possibly different indices of stability between 1 and 2. The suggested estimator is asymptotically normal with unknown parameters. We apply an asymptotically valid bootstrap to construct a confidence region for the mean vector. Furthermore, a simulation study is performed to show that the estimation method is efficient for conducting inference about the mean vector for multivariate heavy-tailed observations. In the second part, we present the asymptotic distribution of M-estimators for parameters in an unstable AR(p) process. The innovations are assumed to be in the domain of attraction of a stable law with index 0 < α ≤ 2. In particular, when the model involves repeated unit roots or conjugate complex unit roots, M- estimators have a higher asymptotic rate of convergence compared to the least square estimators. Moreover, we show that the asymptotic results can be written as Ito stochastic integrals. Finally, the preceding methodologies lead to develop the asymptotic theory of M-estimators for parameters in unstable AR(p) processes with nonzero location parameter. Similar to the preceding cases, we assume that the process is driven by innovations in the domain of attraction of a stable law with index 0 < α ≤ 2. In this thesis, for all models, we also cover the finite variance case (α = 2).

Masters of Commerce / If share price fluctuations follow a simple random walk then it implies that forecasting models based on historical growth rates have little ability to forecast acceptable share price movements over a certain period. The simple random walk description of share price dynamics is obtained when a large number of investors have equal probability to buy or sell based on their own opinion. This simple random walk description of the stock market is in essence the Efficient Market Hypothesis, EMT. EMT is the central concept around which financial modelling is based which includes the Black-Scholes model and other important theoretical underpinnings of capital market theory like mean-variance portfolio selection, arbitrage pricing theory (APT), security market line and capital asset pricing model (CAPM). These theories, which postulates that risk can be reduced to zero sets the foundation for option pricing and is a key component in financial software packages used for pricing and forecasting in the financial industry. The model used by Black and Scholes and other models mentioned above are Gaussian, i.e. they exhibit a random nature. This Gaussian property and the existence of expected returns and continuous time paths (also Gaussian properties) allow the use of stochastic calculus to solve complex Black- Scholes models. However, if the markets are not Gaussian then the idea that risk can be. (educed to zero can lead to a misleading and potentially disastrous sense of security on the financial markets. This study project test the null hypothesis - share prices on the JSE follow a random walk - by means of graphical techniques such as symmetry plots and Quantile-Quantile plots to analyse the test distributions. In both graphical techniques evidence for the rejection of normality was found. Evidenceleading to the rejection of the hypothesis was also found through nonparametric or distribution free methods at a 1% level of significance for Anderson-Darling and Runs test.

Penalized methods are becoming more and more popular in statistical research. This dissertation research covers two major aspects of applications of penalized methods: variable selection and nonparametric function estimation. The following two paragraphs give brief introductions to each of the two topics. Infinite variance autoregressive models are important for modeling heavy-tailed time series. We use a penalty method to conduct model selection for autoregressive models with innovations in the domain of attraction of a stable law indexed by alpha is an element of (0, 2). We show that by combining the least absolute deviation loss function and the adaptive lasso penalty, we can consistently identify the true model. At the same time, the resulting coefficient estimator converges at a rate of n^(?1/alpha) . The proposed approach gives a unified variable selection procedure for both the finite and infinite variance autoregressive models. While automatic smoothing parameter selection for nonparametric function estimation has been extensively researched for independent data, it is much less so for clustered and longitudinal data. Although leave-subject-out cross-validation (CV) has been widely used, its theoretical property is unknown and its minimization is computationally expensive, especially when there are multiple smoothing parameters. By focusing on penalized modeling methods, we show that leave-subject-out CV is optimal in that its minimization is asymptotically equivalent to the minimization of the true loss function. We develop an efficient Newton-type algorithm to compute the smoothing parameters that minimize the CV criterion. Furthermore, we derive one simplification of the leave-subject-out CV, which leads to a more efficient algorithm for selecting the smoothing parameters. We show that the simplified version of CV criteria is asymptotically equivalent to the unsimplified one and thus enjoys the same optimality property. This CV criterion also provides a completely data driven approach to select working covariance structure using generalized estimating equations in longitudinal data analysis. Our results are applicable to additive, linear varying-coefficient, nonlinear models with data from exponential families.