We study the design of portfolios under a minimum risk criterion. The performance of the optimized portfolio relies on the accuracy of the estimated covariance matrix of the portfolio asset returns. For large portfolios, the number of available marke
t returns is often of similar order to the number of assets, so that the sample covariance matrix performs poorly as a covariance estimator. Additionally, financial market data often contain outliers which, if not correctly handled, may further corrupt the covariance estimation. We address these shortcomings by studying the performance of a hybrid covariance matrix estimator based on Tylers robust M-estimator and on Ledoit-Wolfs shrinkage estimator while assuming samples with heavy-tailed distribution. Employing recent results from random matrix theory, we develop a consistent estimator of (a scaled version of) the realized portfolio risk, which is minimized by optimizing online the shrinkage intensity. Our portfolio optimization method is shown via simulations to outperform existing methods both for synthetic and real market data.
A large dimensional characterization of robust M-estimators of covariance (or scatter) is provided under the assumption that the dataset comprises independent (essentially Gaussian) legitimate samples as well as arbitrary deterministic samples, refer
red to as outliers. Building upon recent random matrix advances in the area of robust statistics, we specifically show that the so-called Maronna M-estimator of scatter asymptotically behaves similar to well-known random matrices when the population and sample sizes grow together to infinity. The introduction of outliers leads the robust estimator to behave asymptotically as the weighted sum of the sample outer products, with a constant weight for all legitimate samples and different weights for the outliers. A fine analysis of this structure reveals importantly that the propensity of the M-estimator to attenuate (or enhance) the impact of outliers is mostly dictated by the alignment of the outliers with the inverse population covariance matrix of the legitimate samples. Thus, robust M-estimators can bring substantial benefits over more simplistic estimators such as the per-sample normalized version of the sample covariance matrix, which is not capable of differentiating the outlying samples. The analysis shows that, within the class of Maronnas estimators of scatter, the Huber estimator is most favorable for rejecting outliers. On the contrary, estimators more similar to Tylers scale invariant estimator (often preferred in the literature) run the risk of inadvertently enhancing some outliers.
This paper presents an analytical characterization of the ergodic capacity of amplify-and-forward (AF) MIMO dual-hop relay channels, assuming that the channel state information is available at the destination terminal only. In contrast to prior resul
ts, our expressions apply for arbitrary numbers of antennas and arbitrary relay configurations. We derive an expression for the exact ergodic capacity, simplified closed-form expressions for the high SNR regime, and tight closed-form upper and lower bounds. These results are made possible to employing recent tools from finite-dimensional random matrix theory to derive new closed-form expressions for various statistical properties of the equivalent AF MIMO dual-hop relay channel, such as the distribution of an unordered eigenvalue and certain random determinant properties. Based on the analytical capacity expressions, we investigate the impact of the system and channel characteristics, such as the antenna configuration and the relay power gain. We also demonstrate a number of interesting relationships between the dual-hop AF MIMO relay channel and conventional point-to-point MIMO channels in various asymptotic regimes.
VERITAS is a new atmospheric Cherenkov imaging telescope array to detect very high energy gamma rays above 100 GeV. The array is located in southern Arizona, USA, at an altitude of 1268m above sea level. The array consists of four 12-m telescopes of
Davies-Cotton design and structurally resembling the Whipple 10-m telescope. The four focal plane instruments are equipped with high-resolution (499 pixels) fast photo-multiplier-tube (PMT) cameras covering a 3.5 degree field of view with 0.15 degree pixel separation. Light concentrators reduce the dead-space between PMTs to 25% and shield the PMTs from ambient light. The PMTs are connected to high-speed preamplifiers allowing operation at modest anode current and giving good single photoelectron peaks in situ. Electronics in the focus box provides real-time monitoring of the anode currents for each pixel and ambient environmental conditions. A charge injection subsystem installed in the focus box allows daytime testing of the trigger and data acquisition system by injecting pulses of variable amplitude and length directly into the photomultiplier preamplifiers. A brief description of the full VERITAS focal plane instrument is given in this paper.
