Confidence bands for distribution functions: The law of the iterated logarithm and shape constraints
In the first part I'll present new goodness-of-fit tests and confidence bands for a distribution function. These are based on suitable versions of the Law of the Iterated Logarithm for stochastic processes on the unit interval. It is shown that these procedures share the good power properties of the Berk and Jones (1979) test and Owen's (1995) confidence band in the tail regions while gaining considerably in the central region.
In the second part these confidence bands are combined with bi-log-concavity, a new shape constraint on a distribution function F: Both log(F) and log(1-F) are assumed to be concave functions. A sufficient condition for bi-log-concavity is log-concavity of the density f = F', but the new constraint is much weaker and includes, for instance, distributions with multiple modes. We present some characterizations of bi-log-concavity. Then it is shown that combining this constraint with suitable confidence bands leads to nontrivial confidence regions for various functionals of F such as moments of arbitrary order.
Finally I'll explain briefly how to extend logistic regression models via bi-log-concavity.
This is joint work with Jon A. Wellner (Seattle), Petro Kolesnyk (Bern) and Ralf Wilke (Copenhagen)
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Determinantal point process models and statistical inference
Statistical models and methods for determinantal point processes (DPPs)
seem largely unexplored,
though they possess a number of appealing properties and have been
studied in mathematical physics, combinatorics, and random matrix theory.
We demonstrate that DPPs provide useful models for the description of
spatial point pattern datasets. Such data are usually modelled by Gibbs point processes,
where the likelihood and moment expressions are intractable and simulations
are time consuming. We exploit the appealing probabilistic properties of
DPPs to develop parametric models, where the likelihood and moment expressions can be easily
evaluated and realizations can be quickly simulated.
We discuss how statistical inference is conducted using the likelihood
or moment properties of DPP models, and we provide freely available software for
simulation and statistical inference.
F. Lavancier, J. Møller and E. Rubak (2015). Determinantal point process
models and statistical inference. To appear in Journal of Royal
Statistical Society: Series B (Statistical Methodology).
F. Lavancier, J. Møller and E. Rubak. Determinantal point process
models and statistical inference: Extended version. (61 pages.)
Available at arXiv:1205.4818. To appear as `Online supplementary
materials' to the JRSS B paper above.
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