24,067 research outputs found
On Quadratic g-Evaluations/Expectations and Related Analysis
In this paper we extend the notion of g-evaluation, in particular
g-expectation, to the case where the generator g is allowed to have a quadratic
growth. We show that some important properties of the g-expectations, including
a representation theorem between the generator and the corresponding
g-expectation, and consequently the reverse comparison theorem of quadratic
BSDEs as well as the Jensen inequality, remain true in the quadratic case. Our
main results also include a Doob-Meyer type decomposition, the optional
sampling theorem, and the up-crossing inequality. The results of this paper are
important in the further development of the general quadratic nonlinear
expectations.Comment: 27 page
Optimal classification in sparse Gaussian graphic model
Consider a two-class classification problem where the number of features is
much larger than the sample size. The features are masked by Gaussian noise
with mean zero and covariance matrix , where the precision matrix
is unknown but is presumably sparse. The useful features,
also unknown, are sparse and each contributes weakly (i.e., rare and weak) to
the classification decision. By obtaining a reasonably good estimate of
, we formulate the setting as a linear regression model. We propose a
two-stage classification method where we first select features by the method of
Innovated Thresholding (IT), and then use the retained features and Fisher's
LDA for classification. In this approach, a crucial problem is how to set the
threshold of IT. We approach this problem by adapting the recent innovation of
Higher Criticism Thresholding (HCT). We find that when useful features are rare
and weak, the limiting behavior of HCT is essentially just as good as the
limiting behavior of ideal threshold, the threshold one would choose if the
underlying distribution of the signals is known (if only). Somewhat
surprisingly, when is sufficiently sparse, its off-diagonal
coordinates usually do not have a major influence over the classification
decision. Compared to recent work in the case where is the identity
matrix [Proc. Natl. Acad. Sci. USA 105 (2008) 14790-14795; Philos. Trans. R.
Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009) 4449-4470], the current
setting is much more general, which needs a new approach and much more
sophisticated analysis. One key component of the analysis is the intimate
relationship between HCT and Fisher's separation. Another key component is the
tight large-deviation bounds for empirical processes for data with
unconventional correlation structures, where graph theory on vertex coloring
plays an important role.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1163 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Representation Theorems for Quadratic -Consistent Nonlinear Expectations
In this paper we extend the notion of ``filtration-consistent nonlinear
expectation" (or "-consistent nonlinear expectation") to the case
when it is allowed to be dominated by a -expectation that may have a
quadratic growth. We show that for such a nonlinear expectation many
fundamental properties of a martingale can still make sense, including the
Doob-Meyer type decomposition theorem and the optional sampling theorem. More
importantly, we show that any quadratic -consistent nonlinear
expectation with a certain domination property must be a quadratic
-expectation. The main contribution of this paper is the finding of the
domination condition to replace the one used in all the previous works, which
is no longer valid in the quadratic case. We also show that the representation
generator must be deterministic, continuous, and actually must be of the simple
form
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