On the inverse power index problem

Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called power indices measuring the influence of a voter. We treat the inverse problem: Given an influence vector and a power index, determine a weighted voting game such that the distribution of influence among the voters is as close as possible to the given target value. We present exact algorithms and computational results for the Shapley-Shubik and the (normalized) Banzhaf power index.Comment: 17 pages, 2 figures, 12 table

Stability for the multifrequency inverse medium problem

The solution of a multi-frequency 1d inverse medium problem consists of recovering the refractive index of a medium from measurements of the scattered waves for multiple frequencies. In this paper, rigorous stability estimates are derived when the frequency takes value in a bounded interval.It is showed that the ill-posedness of the inverse medium problem decreases as the width of the frequency interval becomes larger. More precisely, under certain regularity assumptions on the refractive index, the estimates indicate that the power in H\"older stability is an increasing function of the largest value in the frequency band. Finally, a Lipschitz stability estimate is obtained for the observable part of the medium function defined through a truncated trace formula

PRISM: Sparse Recovery of the Primordial Power Spectrum

The primordial power spectrum describes the initial perturbations in the Universe which eventually grew into the large-scale structure we observe today, and thereby provides an indirect probe of inflation or other structure-formation mechanisms. Here, we introduce a new method to estimate this spectrum from the empirical power spectrum of cosmic microwave background (CMB) maps. A sparsity-based linear inversion method, coined \textbf{PRISM}, is presented. This technique leverages a sparsity prior on features in the primordial power spectrum in a wavelet basis to regularise the inverse problem. This non-parametric approach does not assume a strong prior on the shape of the primordial power spectrum, yet is able to correctly reconstruct its global shape as well as localised features. These advantages make this method robust for detecting deviations from the currently favoured scale-invariant spectrum. We investigate the strength of this method on a set of WMAP 9-year simulated data for three types of primordial power spectra: a nearly scale-invariant spectrum, a spectrum with a small running of the spectral index, and a spectrum with a localised feature. This technique proves to easily detect deviations from a pure scale-invariant power spectrum and is suitable for distinguishing between simple models of the inflation. We process the WMAP 9-year data and find no significant departure from a nearly scale-invariant power spectrum with the spectral index $n_s = 0.972$. A high resolution primordial power spectrum can be reconstructed with this technique, where any strong local deviations or small global deviations from a pure scale-invariant spectrum can easily be detected

Computation of power indices

Power indices are a useful tool for studying voting systems in which different members have different numbers of votes. Many international organisations are organised in this way in order to accommodate differences in size of countries, including the system of QMV in the European Union Council of Ministers. A power index is a measure of power based on the idea that a member’s power is his ability to swing a decision by changing the way his vote is cast. This paper addresses the problem of the computation of the two most widely used power indices, the so-called classical power indices, the Banzhaf index (and also the related Coleman indices) and the Shapley-Shubik index. It discusses the various methods that have been proposed in the literature: Direct Enumeration, Monte Carlo Simulation, Generating Functions, Multilinear Extensions Approximation, the Modified MLE Approximation Method. The advantages and disadvantages of the algorithms are discussed including computational complexity. It also describes methods for so called “oceanic games”. The paper also discusses the so-called “inverse problem” of finding what the weights should be given the desired power indices. The method is potentially useful as providing a basis for designing a voting system with a given desired distribution of power among the members, for example, to reflect differences in population or financial contributions. Examples are given from the International Monetary Fund, shareholder voting and the European Union Council

Estimation of semiparametric stochastic frontiers under shape constraints with application to pollution generating technologies

A number of studies have explored the semi- and nonparametric estimation of stochastic frontier models by using kernel regression or other nonparametric smoothing techniques. In contrast to popular deterministic nonparametric estimators, these approaches do not allow one to impose any shape constraints (or regularity conditions) on the frontier function. On the other hand, as many of the previous techniques are based on the nonparametric estimation of the frontier function, the convergence rate of frontier estimators can be sensitive to the number of inputs, which is generally known as “the curse of dimensionality” problem. This paper proposes a new semiparametric approach for stochastic frontier estimation that avoids the curse of dimensionality and allows one to impose shape constraints on the frontier function. Our approach is based on the singleindex model and applies both single-index estimation techniques and shape-constrained nonparametric least squares. In addition to production frontier and technical efficiency estimation, we show how the technique can be used to estimate pollution generating technologies. The new approach is illustrated by an empirical application to the environmental adjusted performance evaluation of U.S. coal-fired electric power plants.stochastic frontier analysis (SFA), nonparametric least squares, single-index model, sliced inverse regression, monotone rank correlation estimator, environmental efficiency

Updated analytical solutions of continuity equation for electron beams precipitation – I. Pure collisional and pure ohmic energy losses

We present updated analytical solutions of continuity equations for power-law beam electrons precipitating in (a) purely collisional losses and (b) purely ohmic losses. The solutions of continuity equation (CE) normalized on electron density presented in Dobranskis & Zharkova are found by method of characteristics eliminating a mistake in the density characteristic pointed out by Emslie et al. The corrected electron beam differential densities (DD) for collisions are shown to have energy spectra with the index of −(γ + 1)/2, coinciding with the one derived from the inverse problem solution by Brown, while being lower by 1/2 than the index of −γ/2 obtained from CE for electron flux. This leads to a decrease of the index of mean electron spectra from −(γ − 2.5) (CE for flux) to −(γ − 2.0) (CE for electron density). The similar method is applied to CE for electrons precipitating in electric field induced by the beam itself. For the first time, the electron energy spectra are calculated for both constant and variable electric fields by using CE for electron density. We derive electron DD for precipitating electrons (moving towards the photosphere, μ = +1) and ‘returning’ electrons (moving towards the corona, μ = −1). The indices of DD energy spectra are reduced from −γ − 1 (CE for flux) to −γ (CE for electron density). While the index of mean electron spectra is increased by 0.5, from −γ + 0.5 (CE for flux) to −γ + 1(CE for electron density). Hard X-ray intensities are also calculated for relativistic cross-section for the updated differential spectra revealing closer resemblance to numerical Fokker–Planck (FP) solutions