Shape Constrained Inference: Open Problems and New Directions

In parametric statistical inference, data are viewed as generated from an unknown distribution that belongs to a ‘small’ class of possible distributions, usually parameterized by a low dimensional Euclidean parameter vector. The class of normal distributions is an example. In nonparametric inference, the class of possible distributions (the model) is much larger. The class of all distributions on the real line is an example of this

Parametric models are rigid and often too simple to model data realistically. On the other hand, fully non-parametric models suffer from other drawbacks, in particular the well-known curse of dimensionality. An alternative, intermediate approach is to consider constrained nonparametric modeling that directly incorporates prior knowledge about certain aspects of the shape of the underlying distribution into the statistical procedure. Such approaches are practically relevant, as there are many problems, e.g. from stereology and survival analysis, where such knowledge on the shape of the underlying distribution arises from the design of the study.

While statistical methods resulting from this approach often are quite intuitive from a practical point of view, the theoretical analysis and the computations involved tend to be challenging and tackling such problems requires novel ideas. This workshop brings together researchers in statistics, working in the area of shape and geometry, providing a forum to advance both the methodological and the algorithmic aspects of shape constrained inference and other statistical methodologies driven by geometric considerations.