418 research outputs found
A large covariance matrix estimator under intermediate spikiness regimes
The present paper concerns large covariance matrix estimation via composite
minimization under the assumption of low rank plus sparse structure. In this
approach, the low rank plus sparse decomposition of the covariance matrix is
recovered by least squares minimization under nuclear norm plus norm
penalization. This paper proposes a new estimator of that family based on an
additional least-squares re-optimization step aimed at un-shrinking the
eigenvalues of the low rank component estimated at the first step. We prove
that such un-shrinkage causes the final estimate to approach the target as
closely as possible in Frobenius norm while recovering exactly the underlying
low rank and sparsity pattern. Consistency is guaranteed when is at least
, provided that the maximum number of non-zeros per
row in the sparse component is with .
Consistent recovery is ensured if the latent eigenvalues scale to ,
, while rank consistency is ensured if .
The resulting estimator is called UNALCE (UNshrunk ALgebraic Covariance
Estimator) and is shown to outperform state of the art estimators, especially
for what concerns fitting properties and sparsity pattern detection. The
effectiveness of UNALCE is highlighted on a real example regarding ECB banking
supervisory data
The Importance of Being Clustered: Uncluttering the Trends of Statistics from 1970 to 2015
In this paper we retrace the recent history of statistics by analyzing all
the papers published in five prestigious statistical journals since 1970,
namely: Annals of Statistics, Biometrika, Journal of the American Statistical
Association, Journal of the Royal Statistical Society, series B and Statistical
Science. The aim is to construct a kind of "taxonomy" of the statistical papers
by organizing and by clustering them in main themes. In this sense being
identified in a cluster means being important enough to be uncluttered in the
vast and interconnected world of the statistical research. Since the main
statistical research topics naturally born, evolve or die during time, we will
also develop a dynamic clustering strategy, where a group in a time period is
allowed to migrate or to merge into different groups in the following one.
Results show that statistics is a very dynamic and evolving science, stimulated
by the rise of new research questions and types of data
A bootstrap test to detect prominent Granger-causalities across frequencies
Granger-causality in the frequency domain is an emerging tool to analyze the
causal relationship between two time series. We propose a bootstrap test on
unconditional and conditional Granger-causality spectra, as well as on their
difference, to catch particularly prominent causality cycles in relative terms.
In particular, we consider a stochastic process derived applying independently
the stationary bootstrap to the original series. Our null hypothesis is that
each causality or causality difference is equal to the median across
frequencies computed on that process. In this way, we are able to disambiguate
causalities which depart significantly from the median one obtained ignoring
the causality structure. Our test shows power one as the process tends to
non-stationarity, thus being more conservative than parametric alternatives. As
an example, we infer about the relationship between money stock and GDP in the
Euro Area via our approach, considering inflation, unemployment and interest
rates as conditioning variables. We point out that during the period 1999-2017
the money stock aggregate M1 had a significant impact on economic output at all
frequencies, while the opposite relationship is significant only at high
frequencies
High‐dimensional regression coefficient estimation by nuclear norm plus l1 norm penalization
We propose a new estimator of the regression coefficients
for a high-dimensional linear regression model, which is de rived by replacing the sample predictor covariance matrix
in the OLS estimator with a different predictor covariance
matrix estimate obtained by a nuclear norm plus l1 norm
penalization. We call the estimator ALCE-reg. We make a
direct theoretical comparison of the expected mean square
error of ALCE-reg with OLS and RIDGE. We show in a sim ulation study that ALCE-reg is particularly effective when
both the dimension and the sample size are large, due to its
ability to find a good compromise between the large bias of
shrinkage estimators (like RIDGE and LASSO) and the large
variance of estimators conditioned by the sample predictor
covariance matrix (like OLS and POET)
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