This dissertation supplements a significant body of knowledge regarding density representations of empirical commodity and financial series. The justification for this focus is the need to narrow the substantial divide between empirical and statistical curves. The method chosen to accomplish this is closely related to the Gram-Charlier and Edgeworth expansions, where empirical higher order moments appear in an asymptotic series as explicit parameters. Unfortunately the reference density function upon which these series have been built limits input data to that which is generated by a near Gaussian process. Within empirical finance, these normal processes are rarely seen. Data are more usually characterised by non-Gaussian type higher order moments.
In response to the fundamental shortcoming of a widely used and accepted theoretical and empirical model, this research introduces Gram-Charlier type asymptotic expansions that are capable of modelling non-Gaussian behaviour. This is accomplished using the Skewed Student-t (SST) density function of Fernandez and Steel (1998). The resulting SST orthogonal polynomials (which we term Burger polynomials) contain the symmetric Hildebrandt polynomials as a limiting case, with a single parameter controlling for the presence of non-zero skewness. In addition, a derivation based upon the Pearson type IV curve and a determination of a symmetric expansion using the skewed representation of Jones and Faddy (2003) is included.
The success and flexibility of the Skewed Student-t Gram-Charlier type series approximation is measured against a variety of known closed-form probability density functions, using four key empirical series. Results are quantified with the mean-squared-error metric. This lends weight to the hypothesis that the fitted density approximations of these empirical series can be improved, either over the whole range considered or within localised areas of interest. In addition, the developed asymmetric Student-t polynomials are applied to a Value at Risk (VaR) framework. This Skewed Student-t modified VaR (Sk-mVaR) exceedance rate risk metric is compared to the textbook VaR and modified- VaR (mVaR) of Favre and Galeano (2001), with results for each Ï‘-probability, defined as Pr(rt+1)≤VaRt+1)=Ï‘, confirming improvement.
These findings have important implications for a wide variety of applications, including: signal processing, seismic inference, mineral characterisation, temperature prediction, and commodity return forecasting. More broadly, the general statistical method highlighted in this work may improve those modelling applications that rely upon continuous random variables.