On the convergence of Maronna's MM-estimators of scatter

In this paper, {we propose an alternative proof for the uniqueness} of Maronna's $M$-estimator of scatter (Maronna, 1976) for $N$ vector observations $\mathbf y_1,...,\mathbf y_N\in\mathbb R^m$ under a mild constraint of linear independence of any subset of $m$ of these vectors. This entails in particular almost sure uniqueness for random vectors $\mathbf y_i$ with a density as long as $N>m$. {This approach allows to establish further relations that demonstrate that a properly normalized Tyler's $M$-estimator of scatter (Tyler, 1987) can be considered as a limit of Maronna's $M$-estimator. More precisely, the contribution is to show that each $M$-estimator converges towards a particular Tyler's $M$-estimator.} These results find important implications in recent works on the large dimensional (random matrix) regime of robust $M$-estimation

Currency Total Return Swaps: Valuation and Risk Factor Analysis

Currency total return swaps (CTRS) are hybrid derivatives instruments that allow to simultaneously hedge against credit and currency risks. We develop a structural credit risk model to evaluate CTRS premia. Empirical test on a sample of 23,005 price observations from 59 underlying issuers yields an average percentage error of around 10%. This indicates that, beyond interest rate risk, firm-specific factors are major drivers of the variations in the valuation of these instruments. Regression analysis of residuals shows that exchange rate determinants account for up to 40% of model pricing errors – indicating that a currency risk premium affects the CTRS price significantly but only marginally, which confirms the prevalence of credit risk in the pricing of CTRS.Credit derivative, credit risk, currency risk

Comfort and pressure profiles of two auto-adjustable positive airway pressure devices: a technical report

AbstractStudy objectives: The purpose of this study was to compare comfort parameters and pressure profiles of the AutoSetTM (Resmed) and the SOMNOsmartTM (Weinmann), two auto-adjustable positive airway pressure (APAP) devices. Setting: The sleep disorders center of a university hospital. Design: A single-blind randomized trial protocol was applied. A split night procedure allowed each patient to be treated in a crossover fashion with both APAP devices during one overnight study. Patients and methods: Fifty consecutive obstructive sleep apnea (OSA) patients were recruited. Each patient filled out an evaluation form for both devices after the study night. Visual analogue scales were used to score four comfort measures. Three CPAP outcomes generated by the devices (P50, P95 and Pmax) were assessed, compared with each other and correlated with the individually predicted CPAP (Ppred). Results: Forty-five males and 5 females, mean age 53.0 years, body mass index 31.0, were included. The mean apnea-hypopnea index was 58.7, the mean arousal index was 54.3. Mean CPAP-compliance before the titration study was 4.9h per night. Comparison of the two devices regarding the effect on the subjective sleep quality parameters showed no differences. The AutoSetTM pressure outcomes correlated significantly better with Ppred in comparison with the SOMNOsmartTM. The P50 and P95 but not the Pmax values were significantly lower in the SOMNOsmart™ as compared with the AutoSetTM (P50: 5.1±1.3 vs 7.1±1.9mbar, P<0.0001; P95: 7.8±3.0 vs 9.6±1.9mbar, P<0.0005; Pmax: 10.0±3.4 vs 10.8±1.8mbar, NS). Conclusion: While the subjective tolerance of the two APAP machines was comparable, these devices were characterized by different pressure profiles. The pressure parameters of the AutoSetTM correlated better with Ppred than those of the SOMNOsmartTM

Convergence and Fluctuations of Regularized Tyler Estimators

This article studies the behavior of regularized Tyler estimators (RTEs) of scatter matrices. The key advantages of these estimators are twofold. First, they guarantee by construction a good conditioning of the estimate and second, being a derivative of robust Tyler estimators, they inherit their robustness properties, notably their resilience to the presence of outliers. Nevertheless, one major problem that poses the use of RTEs in practice is represented by the question of setting the regularization parameter $\rho$. While a high value of $\rho$ is likely to push all the eigenvalues away from zero, it comes at the cost of a larger bias with respect to the population covariance matrix. A deep understanding of the statistics of RTEs is essential to come up with appropriate choices for the regularization parameter. This is not an easy task and might be out of reach, unless one considers asymptotic regimes wherein the number of observations $n$ and/or their size $N$ increase together. First asymptotic results have recently been obtained under the assumption that $N$ and $n$ are large and commensurable. Interestingly, no results concerning the regime of $n$ going to infinity with $N$ fixed exist, even though the investigation of this assumption has usually predated the analysis of the most difficult $N$ and $n$ large case. This motivates our work. In particular, we prove in the present paper that the RTEs converge to a deterministic matrix when $n\to\infty$ with $N$ fixed, which is expressed as a function of the theoretical covariance matrix. We also derive the fluctuations of the RTEs around this deterministic matrix and establish that these fluctuations converge in distribution to a multivariate Gaussian distribution with zero mean and a covariance depending on the population covariance and the parameter $\rho$

Relaxing order basis computation

International audienceThe computation of an order basis (also called sigma basis) is a fundamental tool for linear algebra with polynomial coefficients. Such a computation is one of the key ingredients to provide algorithms which reduce to polynomial matrices multiplication. This has been the case for column reduction or minimal nullspace basis of polynomial matrix over a field. In this poster, we are interested in the application of order basis to compute minimal matrix generators of a linear matrix sequence. In particular, we focus on the linear matrix sequence used in the Block Wiedemann algorithm

THE EVOLUTION OF MORPHOLOGICAL DIVERSITY AND SEXUAL DIMORPHISM IN STICK AND LEAF INSECTS

How has the diversity of life forms come to be? This question is at the core of evolutionary biology and can be addressed at different scales: by studying the processes that drive modifications within populations of organisms generation after generation (microevolution), or by investigating patterns of changes on the tree of life over long periods of time (macroevolution). Understanding the ultimate drivers of morphological diversity eventually entails connecting microevolutionary processes with macroevolutionary patterns. My dissertation investigates the diversification of body and egg form and its drivers in a relatively small but particularly diverse insect order: the stick and leaf insects (Phasmatodea). As masters of camouflage, the 3,400 described species of phasmids are an ideal system to study morphological evolution as they vary tremendously in body morphology, going from long slender branch mimics to wide, flat animals that look exactly like leaves. This remarkable diversity of forms enables phasmids to avoid detection by visually-hunting predators. Even their remarkably diverse hardshelled eggs resemble a wide variety of plant seeds. In addition, males and females of the same species often look very different from each other, with females in extreme cases more than ten times the size of the males. In chapters one, two and three, I investigate the patterns of variation of female body morphology, sexual dimorphism and egg morphology respectively, and potential ecological, life history and biomechanical correlates in a phylogenetic context. I describe repeated convergence towards multiple body forms associated with habitat transitions but find substantial variation in the strength of convergence and underlying evolutionary paths. Then, I show that variation in the extent of sexual dimorphism is best explained by variation in selective pressures acting on males, namely locomotor (flight) performance and male competition (sexual selection). Finally, I show that variation in egg size and shape is driven by variation in life history strategies, mechanical constraints and oviposition strategy. In chapters four, five and six, I investigate the microevolutionary processes behind the primary macroevolutionary forces driving variation in sexual dimorphism. In chapter four, I show in leaf insects (Phyllium philippinicum) that larger males are poor flyers, suggesting that selection for flight performance favors smaller male body sizes in this species, and reinforcing the broader taxonomic findings of chapter two. In chapters five and six, I describe how a change in the mating system of thorny devil stick insects (Eurycantha calcarata) switched the direction of sexual selection and led to the evolution of exceptionally large male body sizes and exaggerated hindleg weapons, confirming the pervasive role of sexual selection in driving variation in male size and sexual dimorphism. Collectively, my research contributes to our understanding of the forces that shape the evolution of morphology in animals and their eggs