Andreas Frost

Geometric Measure Representation of the Log-Normal Distribution

"I am interested in geometric approaches in statistics and its application to data analysis in life sciences."

Log-normal distributions are ubiquitous in the life sciences. Many natural processes produce log-normally distributed data that must be analysed. But for certain statistics the exact distributions in an underlying log-normal population are unknown. So you often have to rely on asymptotic theory.
The questions we focus on are: How can a geometric measure representation be constructed for a log-normally distributed random vector? For which statistics can exact distributions be derived in an underlying log-normal population using this representation?

By means of a geometric measure representation a high-dimensional log-normally distributed random vector can be described by a one-dimensional integral. The distributions of certain statistics can then be deduced from this representation using geometric reasoning.

Geometric measure representations and distributions of certain statistics have already been derived for continuous ln,p – symmetric distributions (the level sets of their densities are ln,p – balls). Analogous to the methodology used in that case a generalized surface measure for the level sets of the log-normal distribution must be established. This is the basis for the derivation of a geometric measure representation. This representation is finally supposed to be used for the deduction of exact distributions of statistics in a log-normal population.

The results of the work can be a basis for the enrichment of statistical data analysis with exact methods which can improve decision making in life sciences for data generated by a log-normal population.