Among Barndorff-Nielsen's early scientific contributions are his work on
exponential families and on the
foundations of statistics, in particular
sufficiency and conditional inference. In 1977 he introduced the
hyperbolic distribution as a mathematical model of the size distribution of
sand grains, formalising heuristic ideas proposed by
Ralph Alger Bagnold. He also derived the larger class of
generalised hyperbolic distributions. These distributions, in particular the
normal-inverse Gaussian (NIG) distribution, have later turned out to be useful in many other areas of science, in particular
turbulence and
finance. The NIG-distribution is now widely used to describe the distribution of
returns from financial assets. In 1984 he produced a short film on the physics of blown sand and the life of the British scientist and explorer Brigadier
Ralph Alger Bagnold. A follow-up to the film was produced in 2011 on the studies of stochastics in the physical sciences carried out by Barndorff-Nielsen and colleagues at the Faculty of Science, Aarhus University by the initiative of the President of the Bernoulli Society for Mathematical Statistics and Probability, Professor Victor Pérez-Abreu. Later Barndorff-Nielsen played a leading role in the application of
differential geometry to investigate statistical models. Another main contribution is his work on
asymptotic methods in statistics, not least his formula for the
conditional distribution of the
maximum likelihood estimator given an
ancillary statistic that generalizes a formula by
Ronald A. Fisher (originally called the p^*-formula, but now known as the Barndorff-Nielsen formula). He has jointly with
David Cox written two influential books on asymptotic techniques in statistics. Since the mid-90s Barndorff-Nielsen has worked on
stochastic models in finance (often with
Neil Shephard) and turbulence, on statistical methods for the analysis of data from experiments in
quantum physics, and has contributed to the theory of
Lévy processes. == Notable honors and positions held ==