7,218 research outputs found
Perturbation bounds and degree of imprecision for uniquely convergent imprecise Markov chains
The effect of perturbations of parameters for uniquely convergent imprecise
Markov chains is studied. We provide the maximal distance between the
distributions of original and perturbed chain and maximal degree of
imprecision, given the imprecision of the initial distribution. The bounds on
the errors and degrees of imprecision are found for the distributions at finite
time steps, and for the stationary distributions as well.Comment: 20 pages, 2 figure
On Sharp Identification Regions for Regression Under Interval Data
The reliable analysis of interval data (coarsened data) is one of the
most promising applications of imprecise probabilities in statistics. If one
refrains from making untestable, and often materially unjustified, strong
assumptions on the coarsening process, then the empirical distribution
of the data is imprecise, and statistical models are, in Manskiās terms,
partially identified. We first elaborate some subtle differences between
two natural ways of handling interval data in the dependent variable of
regression models, distinguishing between two different types of identification
regions, called Sharp Marrow Region (SMR) and Sharp Collection
Region (SCR) here. Focusing on the case of linear regression analysis, we
then derive some fundamental geometrical properties of SMR and SCR,
allowing a comparison of the regions and providing some guidelines for
their canonical construction.
Relying on the algebraic framework of adjunctions of two mappings between
partially ordered sets, we characterize SMR as a right adjoint and
as the monotone kernel of a criterion function based mapping, while SCR
is indeed interpretable as the corresponding monotone hull. Finally we
sketch some ideas on a compromise between SMR and SCR based on a
set-domained loss function.
This paper is an extended version of a shorter paper with the same title,
that is conditionally accepted for publication in the Proceedings of
the Eighth International Symposium on Imprecise Probability: Theories
and Applications. In the present paper we added proofs and the seventh
chapter with a small Monte-Carlo-Illustration, that would have made the
original paper too long
Data and uncertainty in extreme risks - a nonlinear expectations approach
Estimation of tail quantities, such as expected shortfall or Value at Risk,
is a difficult problem. We show how the theory of nonlinear expectations, in
particular the Data-robust expectation introduced in [5], can assist in the
quantification of statistical uncertainty for these problems. However, when we
are in a heavy-tailed context (in particular when our data are described by a
Pareto distribution, as is common in much of extreme value theory), the theory
of [5] is insufficient, and requires an additional regularization step which we
introduce. By asking whether this regularization is possible, we obtain a
qualitative requirement for reliable estimation of tail quantities and risk
measures, in a Pareto setting
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