Fock factorizations, and decompositions of the L2L^2 spaces over general Levy processes

We explicitly construct and study an isometry between the spaces of square integrable functionals of an arbitrary Levy process and a vector-valued Gaussian white noise. In particular, we obtain explicit formulas for this isometry at the level of multiplicative functionals and at the level of orthogonal decompositions, as well as find its kernel. We consider in detail the central special case: the isometry between the $L^2$ spaces over a Poisson process and the corresponding white noise. The key role in our considerations is played by the notion of measure and Hilbert factorizations and related notions of multiplicative and additive functionals and logarithm. The obtained results allow us to introduce a canonical Fock structure (an analogue of the Wiener--Ito decomposition) in the $L^2$ space over an arbitrary Levy process. An application to the representation theory of current groups is considered. An example of a non-Fock factorization is given.Comment: 35 pages; LaTeX; to appear in Russian Math. Survey

A unified approach to structural change tests based on F statistics, OLS residuals, and ML scores

Three classes of structural change tests (or tests for parameter instability) which have been receiving much attention in both the statistics and econometrics communities but have been developed in rather loosely connected lines of research are unified by embedding them into the framework of generalized M-fluctuation tests (Zeileis and Hornik, 2003). These classes are tests based on F statistics (supF, aveF, expF tests), on OLS residuals (OLS-based CUSUM and MOSUM tests) and on maximum likelihood scores (including the Nyblom-Hansen test). We show that (represantives from) these classes are special cases of the generalized M-fluctuation tests, based on the same functional central limit theorem, but employing different functionals for capturing excessive fluctuations. After embedding these tests into the same framework and thus understanding the relationship between these procedures for testing in historical samples, it is shown how the tests can also be extended to a monitoring situation. This is achieved by establishing a general M-fluctuation monitoring procedure and then applying the different functionals corresponding to monitoring with F statistics, OLS residuals and ML scores. In particular, an extension of the supF test to a monitoring scenario is suggested and illustrated on a real-world data set.Series: Research Report Series / Department of Statistics and Mathematic

On the posterior distribution of classes of random means

The study of properties of mean functionals of random probability measures is an important area of research in the theory of Bayesian nonparametric statistics. Many results are now known for random Dirichlet means, but little is known, especially in terms of posterior distributions, for classes of priors beyond the Dirichlet process. In this paper, we consider normalized random measures with independent increments (NRMI's) and mixtures of NRMI. In both cases, we are able to provide exact expressions for the posterior distribution of their means. These general results are then specialized, leading to distributional results for means of two important particular cases of NRMI's and also of the two-parameter Poisson--Dirichlet process.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ200 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm