27,691 research outputs found
The asymptotic distribution of the isotonic regression estimator over a general countable pre-ordered set
We study the isotonic regression estimator over a general countable
pre-ordered set. We obtain the limiting distribution of the estimator and study
its properties. It is proved that, under some general assumptions, the limiting
distribution of the isotonized estimator is given by the concatenation of the
separate isotonic regressions of the certain subvectors of an unrestrecred
estimator's asymptotic distribution. Also, we show that the isotonization
preserves the rate of convergence of the underlying estimator. We apply these
results to the problems of estimation of a bimonotone regression function and
estimation of a bimonotone probability mass function
Random Metric Spaces and Universality
WWe define the notion of a random metric space and prove that with
probability one such a space is isometricto the Urysohn universal metric space.
The main technique is the study of universal and random distance matrices; we
relate the properties of metric (in particulary universal) space to the
properties of distance matrices. We show the link between those questions and
classification of the Polish spaces with measure (Gromov or metric triples) and
with the problem about S_{\infty}-invariant measures in the space of symmetric
matrices. One of the new effects -exsitence in Urysohn space so called
anarchical uniformly distributed sequences. We give examples of other
categories in which the randomness and universality coincide (graph, etc.).Comment: 38 PAGE
Universality of Bayesian mixture predictors
The problem is that of sequential probability forecasting for finite-valued
time series. The data is generated by an unknown probability distribution over
the space of all one-way infinite sequences. It is known that this measure
belongs to a given set C, but the latter is completely arbitrary (uncountably
infinite, without any structure given). The performance is measured with
asymptotic average log loss. In this work it is shown that the minimax
asymptotic performance is always attainable, and it is attained by a convex
combination of a countably many measures from the set C (a Bayesian mixture).
This was previously only known for the case when the best achievable asymptotic
error is 0. This also contrasts previous results that show that in the
non-realizable case all Bayesian mixtures may be suboptimal, while there is a
predictor that achieves the optimal performance
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