7,389 research outputs found
Multidimensional trimming based on projection depth
As estimators of location parameters, univariate trimmed means are well known
for their robustness and efficiency. They can serve as robust alternatives to
the sample mean while possessing high efficiencies at normal as well as
heavy-tailed models. This paper introduces multidimensional trimmed means based
on projection depth induced regions. Robustness of these depth trimmed means is
investigated in terms of the influence function and finite sample breakdown
point. The influence function captures the local robustness whereas the
breakdown point measures the global robustness of estimators. It is found that
the projection depth trimmed means are highly robust locally as well as
globally. Asymptotics of the depth trimmed means are investigated via those of
the directional radius of the depth induced regions. The strong consistency,
asymptotic representation and limiting distribution of the depth trimmed means
are obtained. Relative to the mean and other leading competitors, the depth
trimmed means are highly efficient at normal or symmetric models and
overwhelmingly more efficient when these models are contaminated. Simulation
studies confirm the validity of the asymptotic efficiency results at finite
samples.Comment: Published at http://dx.doi.org/10.1214/009053606000000713 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Solutions modulo of Gauss-Manin differential equations for multidimensional hypergeometric integrals and associated Bethe ansatz
We consider the Gauss-Manin differential equations for hypergeometric
integrals associated with a family of weighted arrangements of hyperplanes
moving parallelly to themselves. We reduce these equations modulo a prime
integer and construct polynomial solutions of the new differential
equations as -analogs of the initial hypergeometric integrals.
In some cases we interpret the -analogs of the hypergeometric integrals as
sums over points of hypersurfaces defined over the finite field . That
interpretation is similar to the interpretation by Yu.I. Manin in [Ma] of the
number of point on an elliptic curve depending on a parameter as a solution of
a classical hypergeometric differential equation.
We discuss the associated Bethe ansatz.Comment: Latex, 19 pages, v2: misprints correcte
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