Asymmetric Weighting Functions Research Articles

The risk function of the goodness-of-fit tests for tail models

This paper contributes to answering a question that is of crucial importance in risk management and extreme value theory: How to select the threshold above which one assumes that the tail of a distribution follows a generalized Pareto distribution. This question has gained increasing attention, particularly in finance institutions, as the recent regulative norms require the assessment of risk at high quantiles. Recent methods answer this question by multiple uses of the standard goodness-of-fit tests. These tests are based on a particular choice of symmetric weighting of the mean square error between the empirical and the fitted tail distributions. Assuming an asymmetric weighting, which rates high quantiles more than small ones, we propose new goodness-of-fit tests and automated threshold selection procedures. We consider a parameterized family of asymmetric weight functions and calculate the corresponding mean square error as a loss function. We then explicitly determine the risk function as the finite sample expected value of the loss function. Finally, the risk function can be used to discuss the question of which symmetric or asymmetric weight function and, thus, which goodness-of-fit test should be used in a new method for determining the threshold value.

Mahalanobis Distance Cross-Correlation for Illumination-Invariant Stereo Matching

A robust similarity measure called the Mahalanobis distance cross-correlation (MDCC) is proposed for illumination-invariant stereo matching, which uses a local color distribution within support windows. It is shown that the Mahalanobis distance between the color itself and the average color is preserved under affine transformation. The MDCC converts pixels within each support window into the Mahalanobis distance transform (MDT) space. The similarity between MDT pairs is then computed using the cross-correlation with an asymmetric weight function based on the Mahalanobis distance. The MDCC considers correlation on cross-color channels, thus providing robustness to affine illumination variation. Experimental results show that the MDCC outperforms state-of-the-art similarity measures in terms of stereo matching for image pairs taken under different illumination conditions.

A weighted cramer-von mises statistic derived from a decomposition of the anderson-darling statistic

Two goodness of fit statistics with asymmetric weight function are derived from a decomposition of the Anderson-Darling statistic, For each one, the asymptotic null distribution is found for a simple null hypothesis and some upper percentties are calculated. The asymptotic power of the tests are obtained for some contiguous alternatives around a normal null hypothesis. The tests allow the user to choose to which tail to give more weight and it is intended to be used for that purpose. Therefore it should be not considered as a competitor of the classical goodness of fit tests.

Steady laminar flow over a downstream-facing step as a critical test case for the upwind Petrov-Galerkin finite element method

Galerkin finite element methods based on symmetric weighting functions give oscillations in the solution of steady Navier-Stokes equations when the Reynolds number is large. To overcome this difficulty Galerkin methods with asymmetric weighting functions can be used. In the present paper a critical test case is calculated with this so-called Petrov-Galerkin method. Comparison between calculations and experiments proofs that the accuracy of the method is good, even for a Reynolds number based on mesh size as high as two hundred.

Solution of transient transport equation using an upstream finite element scheme

An upstream finite element scheme is presented for overcoming the problem of numerical oscillations associated with the transient transport equation for convective dominated or purely convective flow. This scheme differs from the standard Galerkin scheme in that spatial discretization is performed via a general weighted residual technique which employs asymmetric weighting functions and yields upstream weighting of the convective term in the transport equation. The time derivative term of this equation is weighted using the standard basis functions which lead to a consistent mass matrix. The proposed finite element scheme is compared with the conventional upstream finite difference scheme both in one and two dimensions. Numerical results indicate that the new scheme is superior to its finite difference counterpart and is capable of producing reliable solutions for cases which cannot be handled satisfactorily by the standard Galerkin technique.