Dr. Athanasios G. Georgiadis from the School of Computer Science and Statistics, Trinity College Dublin in collaboration with Dr. Galatia Cleanthous from Maynooth University and Prof. Oleg Lepski from Aix–Marseille Université, have published new research focused on adaptive estimation of the L2-norm of probability densities.
Their work has been published in the Annals of Statistics, one of the most highly respected journals in the field of Statistics.
The study has been published in two papers; containing the relevant lower and upper bounds, respectively.
Their research discusses phenomena that are subject to uncertainty can be represented by a random variable X. The values of X would capture the range of possible outcomes for the phenomenon being studied. By modelling the phenomenon based on X, they could use statistical techniques to analyse and understand its behaviour. This is a common approach in finance, engineering, science and beyond.
All the information of the random phenomenon is in turn compressed in its probability density function (briefly referred as a density). The density of X is something like its DNA, and by "unlocking" it, they gained a full understanding of the X and the phenomenon it captures!
Statistics approaches the unknown random variable X using datasets which consist of independent replicates of it.
In the area of Non-parametric Statistics, they tried to minimize the theoretical assumptions to the density (and therefore the unknown phenomenon under study). Non-parametric estimation relies just on generic functional assumptions, and this makes the methods less restrictive and broadly applicable. But then Adaptive Estimation is the key to maximize the generality of the assumptions, consequently the target domain of densities (and finally the phenomena) they estimate.
In their work, they aim in the adaptive estimation of the "magnitude" of a density, which is expressed commonly through the L_2 norm of it. Problems that have been of interest to the community for decades. The first paper contains the lower and the second the upper bounds of this estimation.
The publication in Annals of Statistics highlights the contribution of this work to the field and its potential to support further developments in statistical theory and practice.