Will (1974) treated the perturbation of a Schwarzschild black hole due to a slowly rotating light concentric thin ring by solving the perturbation equations in terms of a multipole expansion of the mass-and-rotation perturbation series. In the Schwarzschild background, his approach can be generalized to the perturbation by a thin disc (which is more relevant astrophysically), but, due to a rather bad convergence properties, the resulting expansions are not suitable for specific (numerical) computations. However, we show that Green's functions represented by the Will's result can be expressed in a closed form (without multipole expansion) which is more useful. In particular, they can be integrated out over the source (thin disc in our case), to yield well converging series both for the gravitational potential and for the dragging angular velocity. The procedure is demonstrated, in the first perturbation order, on the simplest case of a constant-density disc, including physical interpretation of the results in terms of a one-component perfect fluid or a two-component dust on circular orbits about the central black hole. Free parameters are chosen in such a way that the resulting black hole has zero angular momentum but non-zero angular velocity, being just carried along by the dragging effect of the disc.

This paper offers a new method for estimation and forecasting of the volatility of financial time series when the stationarity assumption is violated. Our general, local parametric approach particularly applies to general varying-coefficient parametric models, such as GARCH, whose coefficients may arbitrarily vary with time. Global parametric, smooth transition and change-point models are special cases. The method is based on an adaptive pointwise selection of the largest interval of homogeneity with a given right-end point by a local change-point analysis. We construct locally adaptive estimates that can perform this task and investigate them both from the theoretical point of view and by Monte Carlo simulations. In the particular case of GARCH estimation, the proposed method is applied to stock-index series and is shown to outperform the standard parametric GARCH model. Copyright � 2009 The Author(s). Journal compilation � Royal Economic Society 2009

The panel-data regression models are frequently applied to micro-level data, which often suffer from data contamination, erroneous observations, or unobserved heterogeneity. Despite the adverse effects of outliers on classical estimation methods, there are only a few robust estimation methods available for fixed-effect panel data. Aiming at estimation under weak moment conditions, a new estimation approach based on two different data transformation is proposed. Considering several robust estimation methods applied on the transformed data, we derive the finite-sample, robust, and asymptotic properties of the proposed estimators including their breakdown points and asymptotic distribution. The finite-sample performance of the existing and proposed methods is compared by means of Monte Carlo simulations.

In order to find the perturbation of a Schwarzschild space-time due to a rotating thin disc, we try to adjust the method used by [4] in the case of perturbation by a one-dimensional ring. This involves solution of stationary axisymmetric Einstein's equations in terms of sphericalharmonic expansions whose convergence however turned out questionable in numerical examples. Here we show, analytically, that the series are almost everywhere convergent, but in some regions the convergence is not absolute.

High breakdown-point regression estimators protect against large errors and data contamination. Motivated by some -the least trimmed squares and maximum trimmed likelihood estimators -we propose a general trimmed estimator, which unifies and extends many existing robust procedures. We derive here the consistency and rate of convergence of the proposed general trimmed estimator under mild β-mixing conditions and demonstrate its applicability in nonlinear regression, time series, limited dependent variable models, and panel data.