Asset Allocation with Aversion to Parameter Uncertainty: A Minimax Regression Approach

This paper takes a minimax regression approach to incorporate aversion to parameter uncertainty into the mean-variance model. The uncertainty-averse minimax mean-variance portfolio is obtained by minimizing with respect to the unknown weights the upper bound of the usual quadratic risk function over a fuzzy ellipsoidal set. Beyond the existing approaches, our methodology offers three main advantages: first, the resulting optimal portfolio can be interpreted as a Bayesian mean-variance portfolio with the least favorable prior density, and this result allows for a comprehensive comparison with traditional uncertainty-neutral Bayesian mean-variance portfolios. Second, the minimax mean-variance portfolio has a shrinkage expression, but its performance does not necessarily lie within those of the two reference portfolios. Third, we provide closed form expressions for the standard errors of the minimax mean-variance portfolio weights and statistical significance of the optimal portfolio weights can be easily conducted. Empirical applications show that incorporating aversion to parameter uncertainty leads to more stable optimal portfolios that outperform traditional uncertainty-neutral Bayesian mean-variance portfolios.Asset allocation, estimation error, aversion to uncertainty, min-imax regression, Bayesian mean-variance portfolios, least favorable prior

Infinite horizon control and minimax observer design for linear DAEs

In this paper we construct an infinite horizon minimax state observer for a linear stationary differential-algebraic equation (DAE) with uncertain but bounded input and noisy output. We do not assume regularity or existence of a (unique) solution for any initial state of the DAE. Our approach is based on a generalization of Kalman's duality principle. The latter allows us to transform minimax state estimation problem into a dual control problem for the adjoint DAE: the state estimate in the original problem becomes the control input for the dual problem and the cost function of the latter is, in fact, the worst-case estimation error. Using geometric control theory, we construct an optimal control in the feed-back form and represent it as an output of a stable LTI system. The latter gives the minimax state estimator. In addition, we obtain a solution of infinite-horizon linear quadratic optimal control problem for DAEs.Comment: This is an extended version of the paper which is to appear in the proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, December 10-13, 201

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Bayesian minimax estimation of the normal model with incomplete prior covariance matrix specification

This work addresses the issue of Bayesian robustness in the multivariate normal model when the prior covariance matrix is not completely specified, but rather is described in terms of positive semi-definite bounds. This occurs in situations where, for example, the only prior information available is the bound on the diagonal of the covariance matrix derived from some physical constraints, and that the covariance matrix is positive semi-definite, but otherwise arbitrary. Under the conditional Gamma-minimax principle, previous work by DasGupta and Studden shows that an analytically exact solution is readily available for a special case where the bound difference is a scaled identity. The goal in this work is to consider this problem for general positive definite matrices. The contribution in this paper is a theoretical study of the geometry of the minimax problem. Extension of previous results to a more general case is shown and a practical algorithm that relies on semi-definite programming and the convexity of the minimax formulation is derived. Although the algorithm is numerically exact for up to the bivariate case, its exactness for other cases remains open. Numerical studies demonstrate the accuracy of the proposed algorithm and the robustness of the minimax solution relative to standard and recently proposed methods

Asymptotic minimax risk of predictive density estimation for non-parametric regression

We consider the problem of estimating the predictive density of future observations from a non-parametric regression model. The density estimators are evaluated under Kullback--Leibler divergence and our focus is on establishing the exact asymptotics of minimax risk in the case of Gaussian errors. We derive the convergence rate and constant for minimax risk among Bayesian predictive densities under Gaussian priors and we show that this minimax risk is asymptotically equivalent to that among all density estimators.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ222 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Minimax Emission Computed Tomography using High-Resolution Anatomical Side Information and B-Spline Models

In this paper a minimax methodology is presented for combining information from two imaging modalities having different intrinsic spatial resolutions. The focus application is emission computed tomography (ECT), a low-resolution modality for reconstruction of radionuclide tracer density, when supplemented by high-resolution anatomical boundary information extracted from a magnetic resonance image (MRI) of the same imaging volume. The MRI boundary within the two-dimensional (2-D) slice of interest is parameterized by a closed planar curve. The Cramer-Rao (CR) lower bound is used to analyze estimation errors for different boundary shapes. Under a spatially inhomogeneous Gibbs field model for the tracer density a representation for the minimax MRI-enhanced tracer density estimator is obtained. It is shown that the estimator is asymptotically equivalent to a penalized maximum likelihood (PML) estimator with resolution-selective Gibbs penalty. Quantitative comparisons are presented using the iterative space alternating generalized expectation maximization (SAGE-FM) algorithm to implement the PML estimator with and without minimax weight averaging.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85822/1/Fessler86.pd