On the Existence of Jenkins-Strebel Differentials Using Harmonic Maps from Surfaces to Graphs

We give a new proof of the existence (\cite{HM}, \cite{Ren}) of a Jenkins-Strebel differential $\Phi$ on a Riemann surface \SR with prescribed heights of cylinders by considering the harmonic map from \SR to the leaf space of the vertical foliation of $\Phi$, thought of as a Riemannian graph. The novelty of the argument is that it is essentially Riemannian as well as elementary; moreover, the harmonic maps existence theory on which it relies is classical, due mostly to Morrey (\cite{Mo}).Comment: 8 pages, 2 figures available upon reques

Non-Existence of Geometric Minimal Foliations in Hyperbolic Three-Manifolds

In this paper we show that every three-dimensional closed hyperbolic manifold admits no locally geometric $1$-parameter family of closed minimal surfaces.Comment: Commentarii Mathematici Helvetici, to appea

Nonlinear shrinkage estimation of large-dimensional covariance matrices

Many statistical applications require an estimate of a covariance matrix and/or its inverse. When the matrix dimension is large compared to the sample size, which happens frequently, the sample covariance matrix is known to perform poorly and may suffer from ill-conditioning. There already exists an extensive literature concerning improved estimators in such situations. In the absence of further knowledge about the structure of the true covariance matrix, the most successful approach so far, arguably, has been shrinkage estimation. Shrinking the sample covariance matrix to a multiple of the identity, by taking a weighted average of the two, turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean, while retaining the sample eigenvectors. Our paper extends this approach by considering nonlinear transformations of the sample eigenvalues. We show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work. As demonstrated in extensive Monte Carlo simulations, the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage.Comment: Published in at http://dx.doi.org/10.1214/12-AOS989 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Subsampling inference in threshold autoregressive models

This paper discusses inference in self-exciting threshold autoregressive (SETAR) models. Of main interest is inference for the threshold parameter. It is well-known that the asymptotics of the corresponding estimator depend upon whether the SETAR model is continuous or not. In the continuous case, the limiting distribution is normal and standard inference is possible. In the discontinuous case, the limiting distribution is non-normal and it is not known how to estimate it consistently. We show that valid inference can be drawn by the use of the subsampling method. Moreover, the method can even be extended to situations where the (dis)continuity of the model is unknown. In this case, the inference for the regression parameters of the model also becomes difficult and subsampling can be used again. In addition, we consider an hypothesis test for the continuity of a SETAR model. A simulation study examines small sample performance and an application illustrates how the proposed methodology works in practice.Publicad