1,167,817 research outputs found
Multivariate Bernoulli distribution
In this paper, we consider the multivariate Bernoulli distribution as a model
to estimate the structure of graphs with binary nodes. This distribution is
discussed in the framework of the exponential family, and its statistical
properties regarding independence of the nodes are demonstrated. Importantly
the model can estimate not only the main effects and pairwise interactions
among the nodes but also is capable of modeling higher order interactions,
allowing for the existence of complex clique effects. We compare the
multivariate Bernoulli model with existing graphical inference models - the
Ising model and the multivariate Gaussian model, where only the pairwise
interactions are considered. On the other hand, the multivariate Bernoulli
distribution has an interesting property in that independence and
uncorrelatedness of the component random variables are equivalent. Both the
marginal and conditional distributions of a subset of variables in the
multivariate Bernoulli distribution still follow the multivariate Bernoulli
distribution. Furthermore, the multivariate Bernoulli logistic model is
developed under generalized linear model theory by utilizing the canonical link
function in order to include covariate information on the nodes, edges and
cliques. We also consider variable selection techniques such as LASSO in the
logistic model to impose sparsity structure on the graph. Finally, we discuss
extending the smoothing spline ANOVA approach to the multivariate Bernoulli
logistic model to enable estimation of non-linear effects of the predictor
variables.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP10 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Multivariate Juggling Probabilities
We consider refined versions of Markov chains related to juggling introduced
by Warrington. We further generalize the construction to juggling with
arbitrary heights as well as infinitely many balls, which are expressed more
succinctly in terms of Markov chains on integer partitions. In all cases, we
give explicit product formulas for the stationary probabilities. The
normalization factor in one case can be explicitly written as a homogeneous
symmetric polynomial. We also refine and generalize enriched Markov chains on
set partitions. Lastly, we prove that in one case, the stationary distribution
is attained in bounded time.Comment: 28 pages, 5 figures, final versio
Multivariate volatility models
Correlations between asset returns are important in many financial
applications. In recent years, multivariate volatility models have been used to
describe the time-varying feature of the correlations. However, the curse of
dimensionality quickly becomes an issue as the number of correlations is
for assets. In this paper, we review some of the commonly used
models for multivariate volatility and propose a simple approach that is
parsimonious and satisfies the positive definite constraints of the
time-varying correlation matrix. Real examples are used to demonstrate the
proposed model.Comment: Published at http://dx.doi.org/10.1214/074921706000001058 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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