136,595 research outputs found
Multivariate Krawtchouk polynomials and composition birth and death processes
This paper defines the multivariate Krawtchouk polynomials, orthogonal on the
multinomial distribution, and summarizes their properties as a review. The
multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets
of functions defined on each of N multinomial trials. The dual multivariate
Krawtchouk polynomials, which also have a polynomial structure, are seen to
occur naturally as spectral orthogonal polynomials in a Karlin and McGregor
spectral representation of transition functions in a composition birth and
death process. In this Markov composition process in continuous time there are
N independent and identically distributed birth and death processes each with
support 0,1, .... The state space in the composition process is the number of
processes in the different states 0,1,... Dealing with the spectral
representation requires new extensions of the multivariate Krawtchouk
polynomials to orthogonal polynomials on a multinomial distribution with a
countable infinity of states
The CLT Analogue for Cyclic Urns
A cyclic urn is an urn model for balls of types where in each
draw the ball drawn, say of type , is returned to the urn together with a
new ball of type . The case is the well-known Friedman urn.
The composition vector, i.e., the vector of the numbers of balls of each type
after steps is, after normalization, known to be asymptotically normal for
. For the normalized composition vector does not
converge. However, there is an almost sure approximation by a periodic random
vector. In this paper the asymptotic fluctuations around this periodic random
vector are identified. We show that these fluctuations are asymptotically
normal for all . However, they are of maximal dimension only when
does not divide . For being a multiple of the fluctuations are
supported by a two-dimensional subspace.Comment: Extended abstract to be replaced later by a full versio
Hierarchical spatial models for predicting tree species assemblages across large domains
Spatially explicit data layers of tree species assemblages, referred to as
forest types or forest type groups, are a key component in large-scale
assessments of forest sustainability, biodiversity, timber biomass, carbon
sinks and forest health monitoring. This paper explores the utility of coupling
georeferenced national forest inventory (NFI) data with readily available and
spatially complete environmental predictor variables through spatially-varying
multinomial logistic regression models to predict forest type groups across
large forested landscapes. These models exploit underlying spatial associations
within the NFI plot array and the spatially-varying impact of predictor
variables to improve the accuracy of forest type group predictions. The
richness of these models incurs onerous computational burdens and we discuss
dimension reducing spatial processes that retain the richness in modeling. We
illustrate using NFI data from Michigan, USA, where we provide a comprehensive
analysis of this large study area and demonstrate improved prediction with
associated measures of uncertainty.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS250 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Natural statistics for spectral samples
Spectral sampling is associated with the group of unitary transformations
acting on matrices in much the same way that simple random sampling is
associated with the symmetric group acting on vectors. This parallel extends to
symmetric functions, k-statistics and polykays. We construct spectral
k-statistics as unbiased estimators of cumulants of trace powers of a suitable
random matrix. Moreover we define normalized spectral polykays in such a way
that when the sampling is from an infinite population they return products of
free cumulants.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1107 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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