1,492 research outputs found
Improved estimation of the covariance matrix of stock returns with an application to portofolio selection
This paper proposes to estimate the covariance matrix of stock returns by an optimally weighted average of two existing estimators: the sample covariance matrix and single-index covariance matrix. This method is generally known as shrinkage, and it is standard in decision theory and in empirical Bayesian statistics. Our shrinkage estimator can be seen as a way to account for extra-market covariance without having to specify an arbitrary multi-factor structure. For NYSE and AMEX stock returns from 1972 to 1995, it can be used to select portfolios with significantly lower out-of-sample variance than a set of existing estimators, including multi-factor models.Covariance matrix estimation, factor models, portofolio selection, shrinkage
Exploring Estimator Bias-Variance Tradeoffs Using the Uniform CR Bound
We introduce a plane, which we call the delta-sigma plane, that is indexed by the norm of the estimator bias gradient and the variance of the estimator. The norm of the bias gradient is related to the maximum variation in the estimator bias function over a neighborhood of parameter space. Using a uniform Cramer-Rao (CR) bound on estimator variance, a delta-sigma tradeoff curve is specified that defines an “unachievable region” of the delta-sigma plane for a specified statistical model. In order to place an estimator on this plane for comparison with the delta-sigma tradeoff curve, the estimator variance, bias gradient, and bias gradient norm must be evaluated. We present a simple and accurate method for experimentally determining the bias gradient norm based on applying a bootstrap estimator to a sample mean constructed from the gradient of the log-likelihood. We demonstrate the methods developed in this paper for linear Gaussian and nonlinear Poisson inverse problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86001/1/Fessler98.pd
Structured least squares problems and robust estimators
Cataloged from PDF version of article.A novel approach is proposed to provide robust and
accurate estimates for linear regression problems when both the
measurement vector and the coefficient matrix are structured and
subject to errors or uncertainty. A new analytic formulation is developed
in terms of the gradient flow of the residual norm to analyze
and provide estimates to the regression. The presented analysis
enables us to establish theoretical performance guarantees to compare
with existing methods and also offers a criterion to choose the
regularization parameter autonomously. Theoretical results and
simulations in applications such as blind identification, multiple
frequency estimation and deconvolution show that the proposed
technique outperforms alternative methods in mean-squared error
for a significant range of signal-to-noise ratio values
Slepian functions and their use in signal estimation and spectral analysis
It is a well-known fact that mathematical functions that are timelimited (or
spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the
finite precision of measurement and computation unavoidably bandlimits our
observation and modeling scientific data, and we often only have access to, or
are only interested in, a study area that is temporally or spatially bounded.
In the geosciences we may be interested in spectrally modeling a time series
defined only on a certain interval, or we may want to characterize a specific
geographical area observed using an effectively bandlimited measurement device.
It is clear that analyzing and representing scientific data of this kind will
be facilitated if a basis of functions can be found that are "spatiospectrally"
concentrated, i.e. "localized" in both domains at the same time. Here, we give
a theoretical overview of one particular approach to this "concentration"
problem, as originally proposed for time series by Slepian and coworkers, in
the 1960s. We show how this framework leads to practical algorithms and
statistically performant methods for the analysis of signals and their power
spectra in one and two dimensions, and on the surface of a sphere.Comment: Submitted to the Handbook of Geomathematics, edited by Willi Freeden,
Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verla
Multi-output multilevel best linear unbiased estimators via semidefinite programming
Multifidelity forward uncertainty quantification (UQ) problems often involve
multiple quantities of interest and heterogeneous models (e.g., different
grids, equations, dimensions, physics, surrogate and reduced-order models).
While computational efficiency is key in this context, multi-output strategies
in multilevel/multifidelity methods are either sub-optimal or non-existent. In
this paper we extend multilevel best linear unbiased estimators (MLBLUE) to
multi-output forward UQ problems and we present new semidefinite programming
formulations for their optimal setup. Not only do these formulations yield the
optimal number of samples required, but also the optimal selection of
low-fidelity models to use. While existing MLBLUE approaches are single-output
only and require a non-trivial nonlinear optimization procedure, the new
multi-output formulations can be solved reliably and efficiently. We
demonstrate the efficacy of the new methods and formulations in practical UQ
problems with model heterogeneity.Comment: 22 pages, 5 figures, 3 table
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
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