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On the Anti-Wishart distribution
We provide the probability distribution function of matrix elements each of
which is the inner product of two vectors.
The vectors we are considering here are independently distributed but not
necessarily Gaussian variables.
When the number of components M of each vector is greater than the number of
vectors N, one has a symmetric matrix.
When and the components of each vector are independent Gaussian
variables, the distribution function of the matrix elements was
obtained by Wishart in 1928.
When N > M, what we called the ``Anti-Wishart'' case, the matrix elements are
no longer completely independent because the true degrees of freedom becomes
smaller than the number of matrix elements. Due to this singular nature,
analytical derivation of the probability distribution function is much more
involved than the corresponding Wishart case. For a class of general random
vectors, we obtain the analytical distribution function in a closed form, which
is a product of various factors and delta function constraints, composed of
various determinants. The distribution function of the matrix element for the
case with the same class of random vectors is also obtained as a
by-product. Our result is closely related to and should be valuable for the
study of random magnet problem and information redundancy problem.Comment: to appear in Physica
APPLE: Approximate Path for Penalized Likelihood Estimators
In high-dimensional data analysis, penalized likelihood estimators are shown
to provide superior results in both variable selection and parameter
estimation. A new algorithm, APPLE, is proposed for calculating the Approximate
Path for Penalized Likelihood Estimators. Both the convex penalty (such as
LASSO) and the nonconvex penalty (such as SCAD and MCP) cases are considered.
The APPLE efficiently computes the solution path for the penalized likelihood
estimator using a hybrid of the modified predictor-corrector method and the
coordinate-descent algorithm. APPLE is compared with several well-known
packages via simulation and analysis of two gene expression data sets.Comment: 24 pages, 9 figure
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