28,639 research outputs found
The EM Algorithm and the Rise of Computational Biology
In the past decade computational biology has grown from a cottage industry
with a handful of researchers to an attractive interdisciplinary field,
catching the attention and imagination of many quantitatively-minded
scientists. Of interest to us is the key role played by the EM algorithm during
this transformation. We survey the use of the EM algorithm in a few important
computational biology problems surrounding the "central dogma"; of molecular
biology: from DNA to RNA and then to proteins. Topics of this article include
sequence motif discovery, protein sequence alignment, population genetics,
evolutionary models and mRNA expression microarray data analysis.Comment: Published in at http://dx.doi.org/10.1214/09-STS312 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Large Covariance Estimation by Thresholding Principal Orthogonal Complements
This paper deals with the estimation of a high-dimensional covariance with a
conditional sparsity structure and fast-diverging eigenvalues. By assuming
sparse error covariance matrix in an approximate factor model, we allow for the
presence of some cross-sectional correlation even after taking out common but
unobservable factors. We introduce the Principal Orthogonal complEment
Thresholding (POET) method to explore such an approximate factor structure with
sparsity. The POET estimator includes the sample covariance matrix, the
factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding
estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator
(Cai and Liu, 2011) as specific examples. We provide mathematical insights when
the factor analysis is approximately the same as the principal component
analysis for high-dimensional data. The rates of convergence of the sparse
residual covariance matrix and the conditional sparse covariance matrix are
studied under various norms. It is shown that the impact of estimating the
unknown factors vanishes as the dimensionality increases. The uniform rates of
convergence for the unobserved factors and their factor loadings are derived.
The asymptotic results are also verified by extensive simulation studies.
Finally, a real data application on portfolio allocation is presented
Risks of Large Portfolios
Estimating and assessing the risk of a large portfolio is an important topic
in financial econometrics and risk management. The risk is often estimated by a
substitution of a good estimator of the volatility matrix. However, the
accuracy of such a risk estimator for large portfolios is largely unknown, and
a simple inequality in the previous literature gives an infeasible upper bound
for the estimation error. In addition, numerical studies illustrate that this
upper bound is very crude. In this paper, we propose factor-based risk
estimators under a large amount of assets, and introduce a high-confidence
level upper bound (H-CLUB) to assess the accuracy of the risk estimation. The
H-CLUB is constructed based on three different estimates of the volatility
matrix: sample covariance, approximate factor model with known factors, and
unknown factors (POET, Fan, Liao and Mincheva, 2013). For the first time in the
literature, we derive the limiting distribution of the estimated risks in high
dimensionality. Our numerical results demonstrate that the proposed upper
bounds significantly outperform the traditional crude bounds, and provide
insightful assessment of the estimation of the portfolio risks. In addition,
our simulated results quantify the relative error in the risk estimation, which
is usually negligible using 3-month daily data. Finally, the proposed methods
are applied to an empirical study
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