[Technical series]

The standard way of specifying statistical distributions is via their probability mass or probability density function, or via their distribution function, F(x) = P(X ≤ x).

This seminar is a guide to the distributions that can be defined by the inverse of the distribution function, the quantile function, Q(p), where Q(p)=x where x is the smallest x such that P(X ≤ x)=p.

There are theoretical and computational advantages to the quantile approach, which I will illustrate by introducing a number of different distributions, including the generalised lambda, the quantile defined skew logistic, and the logistic-exponential.

I have been involved with the University of Newcastle's robot soccer team, the NUBots for a number of years. The NUBots compete at the world championships each year. Students from computer science, computer engineering and maths and stats take part.

In this introductory seminar I will explain the ways that statistics is used in robot soccer. Statistics plays a part in localisation, how the robot works out where it is; vision, how the robot identifies the things it sees, and behaviour, how the robot decides what to do. As part of this overview I will introduce the statistical topics of classification, Bayesian belief updating, multivariate distributions, predictive models and cross-validation.

Flexibly shaped distributions can provide a wide variety of shapes within the one distributional form. This makes them useful for data-fitting, modelling and simulation. When we consider distributions via the quantile function, rather than their density or distribution functions, it is possible to see the shape behaviour of some distributions in a richer way.

There are many methods of characterising the properties of a distribution. As well as moments and quantile-based measures of location, scale and shape, there are also L-Moments. L-Moments provide an alternative to moments with the advantage of existing for a wider range of shapes than moments do, and having a easier definition in terms of the quantile function than is the case for moments. This seminar reports on my SSP, including an emphasis on L-Moments for the Generalised Lambda Distributions, with an excursus or two on presentation strategies, social boundaries and R Markdown for reproducible research.