3,700 research outputs found
Large Dimensional Analysis and Optimization of Robust Shrinkage Covariance Matrix Estimators
This article studies two regularized robust estimators of scatter matrices
proposed (and proved to be well defined) in parallel in (Chen et al., 2011) and
(Pascal et al., 2013), based on Tyler's robust M-estimator (Tyler, 1987) and on
Ledoit and Wolf's shrinkage covariance matrix estimator (Ledoit and Wolf,
2004). These hybrid estimators have the advantage of conveying (i) robustness
to outliers or impulsive samples and (ii) small sample size adequacy to the
classical sample covariance matrix estimator. We consider here the case of
i.i.d. elliptical zero mean samples in the regime where both sample and
population sizes are large. We demonstrate that, under this setting, the
estimators under study asymptotically behave similar to well-understood random
matrix models. This characterization allows us to derive optimal shrinkage
strategies to estimate the population scatter matrix, improving significantly
upon the empirical shrinkage method proposed in (Chen et al., 2011).Comment: Journal of Multivariate Analysi
Performance analysis and optimal selection of large mean-variance portfolios under estimation risk
We study the consistency of sample mean-variance portfolios of arbitrarily
high dimension that are based on Bayesian or shrinkage estimation of the input
parameters as well as weighted sampling. In an asymptotic setting where the
number of assets remains comparable in magnitude to the sample size, we provide
a characterization of the estimation risk by providing deterministic
equivalents of the portfolio out-of-sample performance in terms of the
underlying investment scenario. The previous estimates represent a means of
quantifying the amount of risk underestimation and return overestimation of
improved portfolio constructions beyond standard ones. Well-known for the
latter, if not corrected, these deviations lead to inaccurate and overly
optimistic Sharpe-based investment decisions. Our results are based on recent
contributions in the field of random matrix theory. Along with the asymptotic
analysis, the analytical framework allows us to find bias corrections improving
on the achieved out-of-sample performance of typical portfolio constructions.
Some numerical simulations validate our theoretical findings
A Robust Statistics Approach to Minimum Variance Portfolio Optimization
We study the design of portfolios under a minimum risk criterion. The
performance of the optimized portfolio relies on the accuracy of the estimated
covariance matrix of the portfolio asset returns. For large portfolios, the
number of available market returns is often of similar order to the number of
assets, so that the sample covariance matrix performs poorly as a covariance
estimator. Additionally, financial market data often contain outliers which, if
not correctly handled, may further corrupt the covariance estimation. We
address these shortcomings by studying the performance of a hybrid covariance
matrix estimator based on Tyler's robust M-estimator and on Ledoit-Wolf's
shrinkage estimator while assuming samples with heavy-tailed distribution.
Employing recent results from random matrix theory, we develop a consistent
estimator of (a scaled version of) the realized portfolio risk, which is
minimized by optimizing online the shrinkage intensity. Our portfolio
optimization method is shown via simulations to outperform existing methods
both for synthetic and real market data
Shrinkage Estimation of the Power Spectrum Covariance Matrix
We seek to improve estimates of the power spectrum covariance matrix from a
limited number of simulations by employing a novel statistical technique known
as shrinkage estimation. The shrinkage technique optimally combines an
empirical estimate of the covariance with a model (the target) to minimize the
total mean squared error compared to the true underlying covariance. We test
this technique on N-body simulations and evaluate its performance by estimating
cosmological parameters. Using a simple diagonal target, we show that the
shrinkage estimator significantly outperforms both the empirical covariance and
the target individually when using a small number of simulations. We find that
reducing noise in the covariance estimate is essential for properly estimating
the values of cosmological parameters as well as their confidence intervals. We
extend our method to the jackknife covariance estimator and again find
significant improvement, though simulations give better results. Even for
thousands of simulations we still find evidence that our method improves
estimation of the covariance matrix. Because our method is simple, requires
negligible additional numerical effort, and produces superior results, we
always advocate shrinkage estimation for the covariance of the power spectrum
and other large-scale structure measurements when purely theoretical modeling
of the covariance is insufficient.Comment: 9 pages, 7 figures (1 new), MNRAS, accepted. Changes to match
accepted version, including an additional explanatory section with 1 figur
Statistical inference for the EU portfolio in high dimensions
In this paper, using the shrinkage-based approach for portfolio weights and
modern results from random matrix theory we construct an effective procedure
for testing the efficiency of the expected utility (EU) portfolio and discuss
the asymptotic behavior of the proposed test statistic under the
high-dimensional asymptotic regime, namely when the number of assets
increases at the same rate as the sample size such that their ratio
approaches a positive constant as . We provide an
extensive simulation study where the power function and receiver operating
characteristic curves of the test are analyzed. In the empirical study, the
methodology is applied to the returns of S\&P 500 constituents.Comment: 27 pages, 5 figures, 2 table
Cleaning large correlation matrices: tools from random matrix theory
This review covers recent results concerning the estimation of large
covariance matrices using tools from Random Matrix Theory (RMT). We introduce
several RMT methods and analytical techniques, such as the Replica formalism
and Free Probability, with an emphasis on the Marchenko-Pastur equation that
provides information on the resolvent of multiplicatively corrupted noisy
matrices. Special care is devoted to the statistics of the eigenvectors of the
empirical correlation matrix, which turn out to be crucial for many
applications. We show in particular how these results can be used to build
consistent "Rotationally Invariant" estimators (RIE) for large correlation
matrices when there is no prior on the structure of the underlying process. The
last part of this review is dedicated to some real-world applications within
financial markets as a case in point. We establish empirically the efficacy of
the RIE framework, which is found to be superior in this case to all previously
proposed methods. The case of additively (rather than multiplicatively)
corrupted noisy matrices is also dealt with in a special Appendix. Several open
problems and interesting technical developments are discussed throughout the
paper.Comment: 165 pages, article submitted to Physics Report
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