2,885 research outputs found
The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument
We present new algorithms that efficiently approximate the hypergeometric
function of a matrix argument through its expansion as a series of Jack
functions. Our algorithms exploit the combinatorial properties of the Jack
function, and have complexity that is only linear in the size of the matrix.Comment: 14 pages, 3 figure
Noncentral Student distributed LS and IV Estimators
The distribution of the least squares and the instrumental variable estimators of the coefficients in a linear relation is noncentral student when the data are normally distributed around possibly non-constant means. This is the claim of the paper and we show for what definition of a noncentral student density this claim to compact summary of the vast literature is justified. Unfortunately, the definition of the noncentral student density is as complicated as its parent, the noncentral Wishart density. Both are defined in terms of infinite series of zonal polynomials of all orders. We have developed however a recursive online method that generates these polynomials sequentially ad infinitum for bivariate and trivariate densities. The time is here that the practicality of the theory can be widened considerably.
Counting derangements and Nash equilibria
The maximal number of totally mixed Nash equilibria in games of several
players equals the number of block derangements, as proved by McKelvey and
McLennan.On the other hand, counting the derangements is a well studied
problem. The numbers are identified as linearization coefficients for Laguerre
polynomials. MacMahon derived a generating function for them as an application
of his master theorem. This article relates the algebraic, combinatorial and
game-theoretic problems that were not connected before. New recurrence
relations, hypergeometric formulas and asymptotics for the derangement counts
are derived. An upper bound for the total number of all Nash equilibria is
given.Comment: 22 pages, 1 table; Theorem 3.3 adde
Optimal and fast detection of spatial clusters with scan statistics
We consider the detection of multivariate spatial clusters in the Bernoulli
model with locations, where the design distribution has weakly dependent
marginals. The locations are scanned with a rectangular window with sides
parallel to the axes and with varying sizes and aspect ratios. Multivariate
scan statistics pose a statistical problem due to the multiple testing over
many scan windows, as well as a computational problem because statistics have
to be evaluated on many windows. This paper introduces methodology that leads
to both statistically optimal inference and computationally efficient
algorithms. The main difference to the traditional calibration of scan
statistics is the concept of grouping scan windows according to their sizes,
and then applying different critical values to different groups. It is shown
that this calibration of the scan statistic results in optimal inference for
spatial clusters on both small scales and on large scales, as well as in the
case where the cluster lives on one of the marginals. Methodology is introduced
that allows for an efficient approximation of the set of all rectangles while
still guaranteeing the statistical optimality results described above. It is
shown that the resulting scan statistic has a computational complexity that is
almost linear in .Comment: Published in at http://dx.doi.org/10.1214/09-AOS732 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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