Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators

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Shape constrained estimators in inverse regression models with convolution-type operator

In this paper we are concerned with shape restricted estimation in inverse regression problems with convolution-type operator. We use increasing rearrangements to compute increasingand convex estimates from an (in principle arbitrary) unconstrained estimate of the unknown regression function. An advantage of our approach is that it is not necessary that prior shape information is known to be valid on the complete domain of the regression function. Instead, it is sufficient if it holds on some compact interval. A simulation study shows that the shape restricted estimate on the respective interval is significantly less sensitive to moderate undersmoothing than the unconstrained estimate, which substantially improves applicability of estimates based on data-driven bandwidth estimators. Finally, we demonstrate the application of the increasing estimator by the estimation of the luminosity profile of an elliptical galaxy. Here, a major interest is in reconstructing the central peak of the profile, which, due to its small size, requires to select the bandwidth as small as possible. --convexity,increasing rearrangements,image reconstruction,inverse problems,monotonicity,order restricted inference,regression estimation,shape restrictions

Statistical inference for inverse problems

In this paper we study statistical inference for certain inverse problems. We go beyond mere estimation purposes and review and develop the construction of confidence intervals and confidence bands in some inverse problems, including deconvolution and the backward heat equation. Further, we discuss the construction of certain hypothesis tests, in particular concerning the number of local maxima of the unknown function. The methods are illustrated in a case study, where we analyze the distribution of heliocentric escape velocities of galaxies in the Centaurus galaxy cluster, and provide statistical evidence for its bimodality. --Asymptotic normality,confidence interval,deconvolution,heat equation,modality,statistical inference,statistical inverse problem

An empirical study of correlation and volatility changes of stock indices and their impact on risk figures

During world financial crisis it became obvious that classical models of portfolio theory significantly under-estimated risks, especially with regard to stocks. Instabilities of correlations and volatilities, the relevant parameters characterizing risk, led to over-estimation of diversification effects and consequently to under-estimation of risks. In this article, we analyze the relevant risk parameters concerning stocks during different market periods of the previous decade. We show that parameters and risks significantly change with market periods and find that the impact of fluctuations and estimation errors is ten times larger for volatilities than for correlations. Moreover, it turns out that diversification between sectors is more efficient than diversification between countries

Gas Dynamics in the Milky Way: Second Pattern Speed and Large-Scale Morphology

We present new gas flow models for the Milky Way inside the solar circle. To this end we use SPH simulations in gravitational potentials determined from the NIR luminosity distribution (including spiral arms) which are based on the COBE/DIRBE maps. Gas flows in models which include massive spiral arms clearly match the observed 12CO lvplot better than if the potential does not include spiral structure. Besides single pattern speed models we investigate models with separate pattern speeds for the bar and spiral arms. The most important difference is that in the latter case the gas spiral arms go through the bar corotation region, keeping the gas aligned with the arms there. In the (l,v) plot this results in characteristic regions which appear to be nearly void of gas.In single pattern speed models these regions are filled with gas because the spiral arms dissolve in the bar corotation region. Comparing with the 12CO data we find evidence for separate pattern speeds in the Milky Way.From a series of models the preferred range for the bar pattern speed is Om_p=60\pm5 /Gyr, corresponding to corotation at 3.4\pm0.3kpc. The spiral pattern speed is less well constrained, but our preferred value is Om_sp\approx 20 /Gyr. A further series of gas models is computed for different bar angles, using separately determined luminosity models and gravitational potentials in each case. We find acceptable gas models for 20<=\phibar<=25. The model with (\phibar=20, Om_p=60 /Gyr, Om_sp=20 /Gyr) gives an excellent fit to the spiral arm ridges in the observed (l,v) plot.Comment: Paper accepted for publication in MNRAS. The paper contains many figures. These are not included in the version available here to save download time. A full version can be downloaded from http://latour.stochastik.math.uni-goettingen.de/~downloads/sphpaper.ps.g

Diversification effects between stock indices

During the World Financial Crisis it became obvious that classical models of portfolio theory significantly under-estimated risks, especially with regard to stocks. Instabilities of correlations and volatilities, the relevant parameters characterizing risk, led to overestimation of diversification effects and consequently to under-estimation of risks. In this article, we analyze diversification effects concerning stocks during different market periods of the previous decade. We show that parameters and risks significantly change with market periods and find that the impact of fluctuations and estimation errors is 5 times larger for volatilities than for correlations. Moreover, it turns out that diversification between sectors is more efficient than diversification between countries

Smooth backfitting in additive inverse regression

We consider the problem of estimating an additive regression function in an inverse regres- sion model with a convolution type operator. A smooth backfitting procedure is developed and asymptotic normality of the resulting estimator is established. Compared to other meth- ods for the estimation in additive models the new approach neither requires observations on a regular grid nor the estimation of the joint density of the predictor. It is also demonstrated by means of a simulation study that the backfitting estimator outperforms the marginal in- tegration method at least by a factor two with respect to the integrated mean squared error criterion.Comment: Keywords: inverse regression; additive models; curse of dimensionality; smooth backfitting Mathematical subject classification: Primary: 62G20; Secondary 15A29 Pages: 26 Figures:

Monitoring of significant changes over time in fluorescence microscopy imaging of living cells

The question whether structural changes in time-resolved images are of statistical significance, and therefore of scientific interest, or merely emerge from random noise is of great relevance in many practical applications such as live cell uorescence microscopy, where intracellular diffusion processes are investigated. In this paper the statistical recovery of such time-resolved images from fluorescence microscopy of living cells is discussed, based on which a bootstrap method is introduced that allows to both monitor and visualize statistically significant structural changes between individual frames over time. The method can be adopted for use in other imaging systems. It yields a criterion to assess time-resolved small scale structural changes e. g. in the nanometer range. The proposed bootstrap method is based on data reconstruction with a regularization technique as well as new theoretical results on uniform confidence bands for the function of interest in a two-dimensional heteroscedastic nonparametric convolution-type inverse regression model of Poisson-type. Moreover, a data-driven selection method for the regularization parameter based on statistical multiscale methods is discussed. The method can be used for an automatic, data-driven data analysis. The theoretical results are demonstrated in a simulation study and are used to analyze data of fluorescently labeled intracellular transport compartments in living cells