19,500 research outputs found
On multivariate quantiles under partial orders
This paper focuses on generalizing quantiles from the ordering point of view.
We propose the concept of partial quantiles, which are based on a given partial
order. We establish that partial quantiles are equivariant under
order-preserving transformations of the data, robust to outliers, characterize
the probability distribution if the partial order is sufficiently rich,
generalize the concept of efficient frontier, and can measure dispersion from
the partial order perspective. We also study several statistical aspects of
partial quantiles. We provide estimators, associated rates of convergence, and
asymptotic distributions that hold uniformly over a continuum of quantile
indices. Furthermore, we provide procedures that can restore monotonicity
properties that might have been disturbed by estimation error, establish
computational complexity bounds, and point out a concentration of measure
phenomenon (the latter under independence and the componentwise natural order).
Finally, we illustrate the concepts by discussing several theoretical examples
and simulations. Empirical applications to compare intake nutrients within
diets, to evaluate the performance of investment funds, and to study the impact
of policies on tobacco awareness are also presented to illustrate the concepts
and their use.Comment: Published in at http://dx.doi.org/10.1214/10-AOS863 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Regularizing effect and local existence for non-cutoff Boltzmann equation
The Boltzmann equation without Grad's angular cutoff assumption is believed
to have regularizing effect on the solution because of the non-integrable
angular singularity of the cross-section. However, even though so far this has
been justified satisfactorily for the spatially homogeneous Boltzmann equation,
it is still basically unsolved for the spatially inhomogeneous Boltzmann
equation. In this paper, by sharpening the coercivity and upper bound estimates
for the collision operator, establishing the hypo-ellipticity of the Boltzmann
operator based on a generalized version of the uncertainty principle, and
analyzing the commutators between the collision operator and some weighted
pseudo differential operators, we prove the regularizing effect in all (time,
space and velocity) variables on solutions when some mild regularity is imposed
on these solutions. For completeness, we also show that when the initial data
has this mild regularity and Maxwellian type decay in velocity variable, there
exists a unique local solution with the same regularity, so that this solution
enjoys the regularity for positive time
Global existence and full regularity of the Boltzmann equation without angular cutoff
We prove the global existence and uniqueness of classical solutions around an
equilibrium to the Boltzmann equation without angular cutoff in some Sobolev
spaces. In addition, the solutions thus obtained are shown to be non-negative
and in all variables for any positive time. In this paper, we study
the Maxwellian molecule type collision operator with mild singularity. One of
the key observations is the introduction of a new important norm related to the
singular behavior of the cross section in the collision operator. This norm
captures the essential properties of the singularity and yields precisely the
dissipation of the linearized collision operator through the celebrated
H-theorem
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