228,388 research outputs found
Informational Divergence Approximations to Product Distributions
The minimum rate needed to accurately approximate a product distribution
based on an unnormalized informational divergence is shown to be a mutual
information. This result subsumes results of Wyner on common information and
Han-Verd\'{u} on resolvability. The result also extends to cases where the
source distribution is unknown but the entropy is known
Some results on moments and cumulants.
In the present paper we discuss various results related to moments and cumulants of probability distributions and approximations to probability distributions. As the approximations are not necessarily probability distributions themselves, we shall apply the concept of moments and cumulants to more general functions. Recursions are deduced for the moments and cumulants of functions in the form Rka,b as defined by Dhaene & Sundt (1994). We deduce a simple relation between the DePril transform and the cumulants of a function. This relation is appplied to some classes of approximations to probability distributions, in particular the approximations of Hipp and DePril.
The Multivariate Watson Distribution: Maximum-Likelihood Estimation and other Aspects
This paper studies fundamental aspects of modelling data using multivariate
Watson distributions. Although these distributions are natural for modelling
axially symmetric data (i.e., unit vectors where \pm \x are equivalent), for
high-dimensions using them can be difficult. Why so? Largely because for Watson
distributions even basic tasks such as maximum-likelihood are numerically
challenging. To tackle the numerical difficulties some approximations have been
derived---but these are either grossly inaccurate in high-dimensions
(\emph{Directional Statistics}, Mardia & Jupp. 2000) or when reasonably
accurate (\emph{J. Machine Learning Research, W. & C.P., v2}, Bijral \emph{et
al.}, 2007, pp. 35--42), they lack theoretical justification. We derive new
approximations to the maximum-likelihood estimates; our approximations are
theoretically well-defined, numerically accurate, and easy to compute. We build
on our parameter estimation and discuss mixture-modelling with Watson
distributions; here we uncover a hitherto unknown connection to the
"diametrical clustering" algorithm of Dhillon \emph{et al.}
(\emph{Bioinformatics}, 19(13), 2003, pp. 1612--1619).Comment: 24 pages; extensively updated numerical result
Compressing Probability Distributions
We show how to store good approximations of probability distributions in
small space
Power of edge exclusion tests in graphical gaussian models
Asymptotic multivariate normal approximations to the joint distributions of edge exclusion test statistics for saturated graphical Gaussian models are derived. Non-signed and signed square-root versions of the likelihood ratio, Wald and score test statistics are considered. Non-central chi-squared approximations are also considered for the non-signed versions. These approximations are used to estimate the power of edge exclusion tests and an example is presented.<br/
Heterogeneous Basket Options Pricing Using Analytical Approximations
This paper proposes the use of analytical approximations to price an heterogeneous basket option combining commodity prices, foreign currencies and zero-coupon bonds. We examine the performance of three moment matching approximations: inverse gamma, Edgeworth expansion around the lognormal and Johnson family distributions. Since there is no closed-form formula for basket options, we carry out Monte Carlo simulations to generate the benchmark values. We perfom a simulation experiment on a whole set of options based on a random choice of parameters. Our results show that the lognormal and Johnson distributions give the most accurate results.Basket Options, Options Pricing, Analytical Approximations, Monte Carlo Simulation
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