4,242 research outputs found
Continuity of the martingale optimal transport problem on the real line
We show continuity of the martingale optimal transport optimisation problem
as a functional of its marginals. This is achieved via an estimate on the
projection in the nested/causal Wasserstein distance of an arbitrary coupling
on to the set of martingale couplings with the same marginals. As a corollary
we obtain an independent proof of sufficiency of the monotonicity principle
established in [Beiglboeck, M., & Juillet, N. (2016). On a problem of optimal
transport under marginal martingale constraints. Ann. Probab., 44 (2016), no.
1, 42106]. On a problem of optimal transport under marginal martingale
constraints. Ann. Probab., 44 (2016), no. 1, 42-106] for cost functions of
polynomial growth
Covariance Estimation in Elliptical Models with Convex Structure
We address structured covariance estimation in Elliptical distribution. We
assume it is a priori known that the covariance belongs to a given convex set,
e.g., the set of Toeplitz or banded matrices. We consider the General Method of
Moments (GMM) optimization subject to these convex constraints. Unfortunately,
GMM is still non-convex due to objective. Instead, we propose COCA - a convex
relaxation which can be efficiently solved. We prove that the relaxation is
tight in the unconstrained case for a finite number of samples, and in the
constrained case asymptotically. We then illustrate the advantages of COCA in
synthetic simulations with structured Compound Gaussian distributions. In these
examples, COCA outperforms competing methods as Tyler's estimate and its
projection onto a convex set
Joint Covariance Estimation with Mutual Linear Structure
We consider the problem of joint estimation of structured covariance
matrices. Assuming the structure is unknown, estimation is achieved using
heterogeneous training sets. Namely, given groups of measurements coming from
centered populations with different covariances, our aim is to determine the
mutual structure of these covariance matrices and estimate them. Supposing that
the covariances span a low dimensional affine subspace in the space of
symmetric matrices, we develop a new efficient algorithm discovering the
structure and using it to improve the estimation. Our technique is based on the
application of principal component analysis in the matrix space. We also derive
an upper performance bound of the proposed algorithm in the Gaussian scenario
and compare it with the Cramer-Rao lower bound. Numerical simulations are
presented to illustrate the performance benefits of the proposed method
Compressed matched filter for non-Gaussian noise
We consider estimation of a deterministic unknown parameter vector in a
linear model with non-Gaussian noise. In the Gaussian case, dimensionality
reduction via a linear matched filter provides a simple low dimensional
sufficient statistic which can be easily communicated and/or stored for future
inference. Such a statistic is usually unknown in the general non-Gaussian
case. Instead, we propose a hybrid matched filter coupled with a randomized
compressed sensing procedure, which together create a low dimensional
statistic. We also derive a complementary algorithm for robust reconstruction
given this statistic. Our recovery method is based on the fast iterative
shrinkage and thresholding algorithm which is used for outlier rejection given
the compressed data. We demonstrate the advantages of the proposed framework
using synthetic simulations
Robust estimation of superhedging prices
We consider statistical estimation of superhedging prices using historical
stock returns in a frictionless market with d traded assets. We introduce a
plugin estimator based on empirical measures and show it is consistent but
lacks suitable robustness. To address this we propose novel estimators which
use a larger set of martingale measures defined through a tradeoff between the
radius of Wasserstein balls around the empirical measure and the allowed norm
of martingale densities. We establish consistency and robustness of these
estimators and argue that they offer a superior performance relative to the
plugin estimator. We generalise the results by replacing the superhedging
criterion with acceptance relative to a risk measure. We further extend our
study, in part, to the case of markets with traded options, to a multiperiod
setting and to settings with model uncertainty. We also study convergence rates
of estimators and convergence of superhedging strategies.Comment: This work will appear in the Annals of Statistics. The above version
merges the main paper to appear in print and its online supplemen
Group Symmetry and non-Gaussian Covariance Estimation
We consider robust covariance estimation with group symmetry constraints.
Non-Gaussian covariance estimation, e.g., Tyler scatter estimator and
Multivariate Generalized Gaussian distribution methods, usually involve
non-convex minimization problems. Recently, it was shown that the underlying
principle behind their success is an extended form of convexity over the
geodesics in the manifold of positive definite matrices. A modern approach to
improve estimation accuracy is to exploit prior knowledge via additional
constraints, e.g., restricting the attention to specific classes of covariances
which adhere to prior symmetry structures. In this paper, we prove that such
group symmetry constraints are also geodesically convex and can therefore be
incorporated into various non-Gaussian covariance estimators. Practical
examples of such sets include: circulant, persymmetric and complex/quaternion
proper structures. We provide a simple numerical technique for finding maximum
likelihood estimates under such constraints, and demonstrate their performance
advantage using synthetic experiments
Tyler's Covariance Matrix Estimator in Elliptical Models with Convex Structure
We address structured covariance estimation in elliptical distributions by
assuming that the covariance is a priori known to belong to a given convex set,
e.g., the set of Toeplitz or banded matrices. We consider the General Method of
Moments (GMM) optimization applied to robust Tyler's scatter M-estimator
subject to these convex constraints. Unfortunately, GMM turns out to be
non-convex due to the objective. Instead, we propose a new COCA estimator - a
convex relaxation which can be efficiently solved. We prove that the relaxation
is tight in the unconstrained case for a finite number of samples, and in the
constrained case asymptotically. We then illustrate the advantages of COCA in
synthetic simulations with structured compound Gaussian distributions. In these
examples, COCA outperforms competing methods such as Tyler's estimator and its
projection onto the structure set.Comment: arXiv admin note: text overlap with arXiv:1311.059
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