Hodges-Lehmann Optimality for Testing Moment

This paper studies the Hodges and Lehmann (1956) optimality of tests in a general setup. The tests are compared by the exponential rates of growth to one of the power functions evaluated at a fixed alternative while keeping the asymptotic sizes bounded by some constant. We present two sets of sufficient conditions for a test to be Hodges-Lehmann optimal. These new conditions extend the scope of the Hodges-Lehmann optimality analysis to setups that cannot be covered by other conditions in the literature. The general result is illustrated by our applications of interest: testing for moment conditions and overidentifying restrictions. In particular, we show that (i) the empirical likelihood test does not necessarily satisfy existing conditions for optimality but does satisfy our new conditions; and (ii) the generalized method of moments (GMM) test and the generalized empirical likelihood (GEL) tests are Hodges-Lehmann optimal under mild primitive conditions. These results support the belief that the Hodges-Lehmann optimality is a weak asymptotic requirement.Asymptotic optimality, Large deviations, Moment condition, Generalized method of moments, Generalized empirical likelihood

Brief history of the Lehmann Symposia: Origins, goals and motivation

The idea of the Lehmann Symposia as platforms to encourage a revival of interest in fundamental questions in theoretical statistics, while keeping in focus issues that arise in contemporary interdisciplinary cutting-edge scientific problems, developed during a conversation that I had with Victor Perez Abreu during one of my visits to Centro de Investigaci\'{o}n en Matem\'{a}ticas (CIMAT) in Guanajuato, Mexico. Our goal was and has been to showcase relevant theoretical work to encourage young researchers and students to engage in such work. The First Lehmann Symposium on Optimality took place in May of 2002 at Centro de Investigaci\'{o}n en Matem\'{a}ticas in Guanajuato, Mexico. A brief account of the Symposium has appeared in Vol. 44 of the Institute of Mathematical Statistics series of Lecture Notes and Monographs. The volume also contains several works presented during the First Lehmann Symposium. All papers were refereed. The program and a picture of the participants can be found on-line at the website http://www.stat.rice.edu/lehmann/lst-Lehmann.html.Comment: Published at http://dx.doi.org/10.1214/074921706000000347 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Spectral representation of lattice gluon and ghost propagators at zero temperature

We consider the analytic continuation of Euclidean propagator data obtained from 4D simulations to Minkowski space. In order to perform this continuation, the common approach is to first extract the K\"all\'en-Lehmann spectral density of the field. Once this is known, it can be extended to Minkowski space to yield the Minkowski propagator. However, obtaining the K\"all\'en-Lehmann spectral density from propagator data is a well known ill-posed numerical problem. To regularize this problem we implement an appropriate version of Tikhonov regularization supplemented with the Morozov discrepancy principle. We will then apply this to various toy model data to demonstrate the conditions of validity for this method, and finally to zero temperature gluon and ghost lattice QCD data. We carefully explain how to deal with the IR singularity of the massless ghost propagator. We also uncover the numerically different performance when using two ---mathematically equivalent--- versions of the K\"all\'en-Lehmann spectral integral.Comment: 33 pages, 18 figure

Dynamics of cholesteric structures in an electric field

Motivated by Lehmann-like rotation phenomena in cholesteric drops we study the transverse drift of two types of cholesteric fingers, which form rotating spirals in thin layers of cholesteric liquid crystal in an ac or dc electric field. We show that electrohydrodynamic effects induced by Carr-Helfrich charge separation or flexoelectric charge generation can describe the drift of cholesteric fingers. We argue that the observed Lehmann-like phenomena can be understood on the same basis.Comment: 4 pages, 4 figures, submitted to PR

Supersymmetric K\"allen-Lehmann representation

We find the general form of supersymmetric invariant two point functions. By imposing supersymmetric positivity we obtain the general supersymmetric K\"allen-Lehmann representation.Comment: 9 page