Nonparametric Finance

Chapter 3Univariate Data Analysis

Univariate data analysis studies univariate financial time series, but ignoring the time series properties of data. Univariate data analysis studies also cross-sectional data. For example, returns at a fixed time point of a collection of stocks is a cross-sectional univariate data set.

A univariate series of observations can be described using such statistics as sample mean, median, variance, quantiles, and expected shortfalls. These are covered in Section 3.1.

The graphical methods are explained in Section 3.2. Univariate graphical tools include tail plots, regression plots of the tails, histograms, and kernel density estimators. We use often tail plots to visualize the tail parts of the distribution, and kernel density estimates to visualize the central part of the distribution. The kernel density estimator is not only a visualization tool but also a tool for estimation.

We define univariate parametric models like normal, log-normal, and Student models in Section 3.3. These are parametric models, which are alternatives to the use of the kernel density estimator.

For a univariate financial time series it is of interest to study the tail properties of the distribution. This is done in Section 3.4. Typically the distribution of a financial time series has heavier tails than the normal distributions. The estimation of the tails is done using the concept of the excess distribution. The excess distribution is modeled with exponential, Pareto, gamma, ...