Band 24, Heft 2

The problem of graph inference, or graph reconstruction, is to predict the presence or absence of edges between a set of given points known to form the vertices of a graph. Motivated by various applications including communication networks and systems biology, we propose a general model for studying the problem of graph inference in a supervised learning framework. In our setting, both the graph vertices and edges are assumed to be random, with a probability distribution that possibly depends on the size of the graph. We show that the problem can be transformed into one where we can use statistical learning methods based on empirical minimization of natural estimates of the reconstruction risk.Convex risk minimizationmethods are also studied to provide a theoretical framework for reconstruction algorithms based on boosting and support vector machines. Our approach is illustrated on simulated graphs.

We consider stochastic volatility models for discrete financial time series of the nonlinear autoregressive-ARCH type with exogenous components.We discuss how the trend and volatility functions determining the process may be estimated nonparametrically by least-squares fitting of neural networks or, more generally, of functions from other parametric classes having a universal approximation property. We prove consistency of the estimates under conditions on the rate of increase of function complexity. The procedure is applied to the problem of quantifying market risk, i.e. of calculating volatility or value-at-risk from the data taking not only the time series of interest but additional market information into account. As an application, we study some stock prices series and compare our approach with the common method based on fitting a GARCH(1,1)-model to the data

In this paper a bond market model and the related term structure of interest rates are studied where prices of zero coupon bonds are driven by a jump-diffusion process. A criterion is derived on the deterministic forward rate volatilities underwhich the short rate process isMarkovian. In the case that the volatilities depend on the short rate sufficient conditions are presented for the existence of a finite-dimensional Markovian realization of the term structure model.

Estimators of multivariate location parameters are generally dominated, in finite as well as asymptotic setups, by suitable shrinkage versions, and hence are inadmissible; such shrinkage estimators may not be admissible either. This feature is shared by maximum likelihood and many robust estimators. The interplay of robustness, admissibility and shrinkage phenomenon in some general multivariate location models (not necessarily elliptically or spherically symmetric) is illustrated and applied to Huber-type contamination models.

Under mild regularity conditions, it is shown that bandwidth selection by minimizing the bootstrapped mean squared error (at a point x ) leads to a bandwidth of the same form as that obtained by a consistent plug-in procedure. The consequences of this observation for the construction of confidence intervals are also discussed.