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Statistical distribution theory with applications to finance
[Thesis]. Manchester, UK: The University of Manchester; 2018.
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Abstract
The whole thesis comprises six chapters, where the running theme focuses on the development of statistical methods and distribution theory, with applications to finance. It begins with Chapter 1, which provides the introduction and background to my thesis. This is then followed by Chapters 2 through to 6, which provide the main contributions. The exact distribution of the sum of more than two independent beta random variables is not a known result. Even in terms of approximations, only the normal approximation is known for the sum. Motivated by Murakami (2014), Chapter 2 derives a saddlepoint approximation for the distribution of the sum. An extensive simulation study shows that it always gives better performance than the normal approximation. Jin et al. (2016) proposed a novel moments based approximation based on the gamma distribution for the compound sum of independent and identical random variables, and illustrated their approximation through the use of six examples. Chapter 3 revisits four of their examples, and it is shown that moments based approximations based on simpler distributions can be good competitors. The moments based approximations are also shown to be more accurate than the truncated versions of the exact distribution of the compound sum. Results regarding the performances of the approximations are provided, which could be useful in determining which approximation should be used given a real data set. The estimation of the size of large populations can often be a significant problem. Chapter 4 proposes a new population size estimator and provides a comparison of its performance with two recent estimators known in the literature. The comparison is based on a simulation study and applications to two real big data sets from the Twitter and LiveJournal social networks. The proposed estimator is shown to outperform the known estimators, at least for small sample sizes. In recent years, with a growing interest in big or large datasets, there has been a rise in the application of large graphs and networks to financial big data. Much of this research has focused on the construction and analysis of the network structure of stock markets, based on the relationships between stock prices. Motivated by Boginski et al. (2005), who studied the characteristics of a network structure of the US stock market, Chapter 5 constructs network graphs of the UK stock market using the same method. Four distributions are fitted to the degree density of the vertices from these graphs: the Pareto I, Frechet, lognormal, and generalised Pareto distributions, and their goodness of fits are assessed. Results show that the degree density of the complements of the market graphs, constructed using a negative threshold value close to zero, can be fitted well with the Frechet and lognormal distributions. Chapter 6 analyses statistical properties of the largest cryptocurrencies (determined by market capitalisation), of which Bitcoin is the most prominent example. The analysis characterises their exchange rates versus the US Dollar by fitting parametric distributions to them. It is shown that cryptocurrency returns are clearly non-normal, however, no single distribution fits well jointly to all of the cryptocurrencies analysed. We find that for the most popular cryptocurrencies, such as Bitcoin and Litecoin, the generalised hyperbolic distribution gives the best fit, whilst for the smaller cryptocurrencies the normal inverse Gaussian distribution, generalised t distribution, and Laplace distribution give good fits. The results are important for investment and risk management purposes.