Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points

It is known that the Fresnel wave surfaces of transparent biaxial media have 4 singular points, located on two special directions. We show that, in more general media, the number of singularities can exceed 4. In fact, a highly symmetric linear material is proposed whose Fresnel surface exhibits 16 singular points. Because, for every linear material, the dispersion equation is quartic, we conclude that 16 is the maximum number of isolated singularities. The identity of Fresnel and Kummer surfaces, which holds true for media with a certain symmetry (zero skewon piece), provides an elegant interpretation of the results. We describe a metamaterial realization for our linear medium with 16 singular points. It is found that an appropriate combination of metal bars, split-ring resonators, and magnetized particles can generate the correct permittivity, permeability, and magnetoelectric moduli. Lastly, we discuss the arrangement of the singularities in terms of Kummer's (16,6)-configuration of points and planes. An investigation parallel to ours, but in linear elasticity, is suggested for future research.Comment: 7 pages, 3 figure

Simple and Bias-Corrected Matching Estimators for Average Treatment Effects

Matching estimators for average treatment effects are widely used in evaluation research despite the fact that their large sample properties have not been established in many cases. In this article, we develop a new framework to analyze the properties of matching estimators and establish a number of new results. First, we show that matching estimators include a conditional bias term which may not vanish at a rate faster than root-N when more than one continuous variable is used for matching. As a result, matching estimators may not be root-N-consistent. Second, we show that even after removing the conditional bias, matching estimators with a fixed number of matches do not reach the semiparametric efficiency bound for average treatment effects, although the efficiency loss may be small. Third, we propose a bias-correction that removes the conditional bias asymptotically, making matching estimators root-N-consistent. Fourth, we provide a new estimator for the conditional variance that does not require consistent nonparametric estimation of unknown functions. We apply the bias-corrected matching estimators to the study of the effects of a labor market program previously analyzed by Lalonde (1986). We also carry out a small simulation study based on Lalonde's example where a simple implementation of the biascorrected matching estimator performs well compared to both simple matching estimators and to regression estimators in terms of bias and root-mean-squared-error. Software for implementing the proposed estimators in STATA and Matlab is available from the authors on the web.

A Martingale Representation for Matching Estimators

Matching estimators are widely used in statistical data analysis. However, the distribution of matching estimators has been derived only for particular cases (Abadie and Imbens, 2006). This article establishes a martingale representation for matching estimators. This representation allows the use of martingale limit theorems to derive the asymptotic distribution of matching estimators. As an illustration of the applicability of the theory, we derive the asymptotic distribution of a matching estimator when matching is carried out without replacement, a result previously unavailable in the literature.matching, martingales, treatment effects, hot-deck imputation

On the Failure of the Bootstrap for Matching Estimators

Matching estimators are widely used for the evaluation of programs or treatments. Often researchers use bootstrapping methods for inference. However, no formal justification for the use of the bootstrap has been provided. Here we show that the bootstrap is in general not valid, even in the simple case with a single continuous covariate when the estimator is root-N consistent and asymptotically normally distributed with zero asymptotic bias. Due to the extreme non-smoothness of nearest neighbor matching, the standard conditions for the bootstrap are not satisfied, leading the bootstrap variance to diverge from the actual variance. Simulations confirm the difference between actual and nominal coverage rates for bootstrap confidence intervals predicted by the theoretical calculations. To our knowledge, this is the first example of a root-N consistent and asymptotically normal estimator for which the bootstrap fails to work.

The Kummer tensor density in electrodynamics and in gravity

Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, ${\cal K}^{ijkl}$. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four ${\cal T}^{ijkl}$, which is antisymmetric in its first two and its last two indices: ${\cal T}^{ijkl} = - {\cal T}^{jikl} = - {\cal T}^{ijlk}$. Thus, ${\cal K}\sim {\cal T}^3$, see Eq.(46). (i) If $\cal T$ is identified with the electromagnetic response tensor of local and linear media, the Kummer tensor density encompasses the generalized {\it Fresnel wave surfaces} for propagating light. In the reversible case, the wave surfaces turn out to be {\it Kummer surfaces} as defined in algebraic geometry (Bateman 1910). (ii) If $\cal T$ is identified with the {\it curvature} tensor $R^{ijkl}$ of a Riemann-Cartan spacetime, then ${\cal K}\sim R^3$ and, in the special case of general relativity, ${\cal K}$ reduces to the Kummer tensor of Zund (1969). This $\cal K$ is related to the {\it principal null directions} of the curvature. We discuss the properties of the general Kummer tensor density. In particular, we decompose $\cal K$ irreducibly under the 4-dimensional linear group $GL(4,R)$ and, subsequently, under the Lorentz group $SO(1,3)$.Comment: 54 pages, 6 figures, written in LaTex; improved version in accordance with the referee repor

Reducing Bias from Choice Experiments Estimates in the Demand for Recreation

In valuing the demand for recreation, the literature has grown from using revealed preference methods to applying stated preference methods, namely contingent valuation and choice modelling. Recent attempts have merged revealed and stated preference data to exploit the strengths of both sources of data. We use contingent behaviour and choice experiments data to show that, with choice experiments exercises, when respondents are asked to choose which improvement programme they prefer for a site with recreational opportunities, failing to consider the information explaining the number of visits that respondents intend to take to a recreational site under each hypothetical programme leads to biased coefficients estimates in the models for the choice experiments data.travel cost, contingent behaviour, choice experiments, revealed preferences, stated preferences, Environmental Economics and Policy, Q51, Q26,