On the long-time behavior of spin echo and its relation to free induction decay

It is predicted that (i) spin echoes have two kinds of generic long-time decays: either simple exponential, or a superposition of a monotonic and an oscillatory exponential decays; and (ii) the long-time behavior of spin echo and the long-time behavior of the corresponding homogeneous free induction decay are characterized by the same time constants. This prediction extends to various echo problems both within and beyond nuclear magnetic resonance. Experimental confirmation of this prediction would also support the notion of the eigenvalues of time evolution operators in large quantum systems.Comment: 4 pages, 4 figure

On asymptotically optimal tests under loss of identifiability in semiparametric models

We consider tests of hypotheses when the parameters are not identifiable under the null in semiparametric models, where regularity conditions for profile likelihood theory fail. Exponential average tests based on integrated profile likelihood are constructed and shown to be asymptotically optimal under a weighted average power criterion with respect to a prior on the nonidentifiable aspect of the model. These results extend existing results for parametric models, which involve more restrictive assumptions on the form of the alternative than do our results. Moreover, the proposed tests accommodate models with infinite dimensional nuisance parameters which either may not be identifiable or may not be estimable at the usual parametric rate. Examples include tests of the presence of a change-point in the Cox model with current status data and tests of regression parameters in odds-rate models with right censored data. Optimal tests have not previously been studied for these scenarios. We study the asymptotic distribution of the proposed tests under the null, fixed contiguous alternatives and random contiguous alternatives. We also propose a weighted bootstrap procedure for computing the critical values of the test statistics. The optimal tests perform well in simulation studies, where they may exhibit improved power over alternative tests.Comment: Published in at http://dx.doi.org/10.1214/08-AOS643 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robust Inference for Univariate Proportional Hazards Frailty Regression Models

We consider a class of semiparametric regression models which are one-parameter extensions of the Cox [J. Roy. Statist. Soc. Ser. B 34 (1972) 187-220] model for right-censored univariate failure times. These models assume that the hazard given the covariates and a random frailty unique to each individual has the proportional hazards form multiplied by the frailty. The frailty is assumed to have mean 1 within a known one-parameter family of distributions. Inference is based on a nonparametric likelihood. The behavior of the likelihood maximizer is studied under general conditions where the fitted model may be misspecified. The joint estimator of the regression and frailty parameters as well as the baseline hazard is shown to be uniformly consistent for the pseudo-value maximizing the asymptotic limit of the likelihood. Appropriately standardized, the estimator converges weakly to a Gaussian process. When the model is correctly specified, the procedure is semiparametric efficient, achieving the semiparametric information bound for all parameter components. It is also proved that the bootstrap gives valid inferences for all parameters, even under misspecification. We demonstrate analytically the importance of the robust inference in several examples. In a randomized clinical trial, a valid test of the treatment effect is possible when other prognostic factors and the frailty distribution are both misspecified. Under certain conditions on the covariates, the ratios of the regression parameters are still identifiable. The practical utility of the procedure is illustrated on a non-Hodgkin's lymphoma dataset.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Statistics (http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000053

Typical state of an isolated quantum system with fixed energy and unrestricted participation of eigenstates

This work describes the statistics for the occupation numbers of quantum levels in a large isolated quantum system, where all possible superpositions of eigenstates are allowed, provided all these superpositions have the same fixed energy. Such a condition is not equivalent to the conventional micro-canonical condition, because the latter limits the participating eigenstates to a very narrow energy window. The statistics is obtained analytically for both the entire system and its small subsystem. In a significant departure from the Boltzmann-Gibbs statistics, the average occupation numbers of quantum states exhibit in the present case weak algebraic dependence on energy. In the macroscopic limit, this dependence is routinely accompanied by the condensation into the lowest energy quantum state. This work contains initial numerical tests of the above statistics for finite systems, and also reports the following numerical finding: When the basis states of large but finite random matrix Hamiltonians are expanded in terms of eigenstates, the participation of eigenstates in such an expansion obeys the newly obtained statistics. The above statistics might be observable in small quantum systems, but for the macroscopic systems, it rather reenforces doubts about self-sufficiency of non-relativistic quantum mechanics for justifying the Boltzmann-Gibbs equilibrium.Comment: 20 pages, 3 figure

Civil Procedure—Specific Exceptions to Charge Needed to Raise Points on Appeal

Lobalzo v. Varoli, 422 Pa. 5, 220 A.2d 634 (1966)

Nonparametric Bounds and Sensitivity Analysis of Treatment Effects

This paper considers conducting inference about the effect of a treatment (or exposure) on an outcome of interest. In the ideal setting where treatment is assigned randomly, under certain assumptions the treatment effect is identifiable from the observable data and inference is straightforward. However, in other settings such as observational studies or randomized trials with noncompliance, the treatment effect is no longer identifiable without relying on untestable assumptions. Nonetheless, the observable data often do provide some information about the effect of treatment, that is, the parameter of interest is partially identifiable. Two approaches are often employed in this setting: (i) bounds are derived for the treatment effect under minimal assumptions, or (ii) additional untestable assumptions are invoked that render the treatment effect identifiable and then sensitivity analysis is conducted to assess how inference about the treatment effect changes as the untestable assumptions are varied. Approaches (i) and (ii) are considered in various settings, including assessing principal strata effects, direct and indirect effects and effects of time-varying exposures. Methods for drawing formal inference about partially identified parameters are also discussed.Comment: Published in at http://dx.doi.org/10.1214/14-STS499 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Phase relationship between the long-time beats of free induction decays and spin echoes in solids

Recent theoretical work on the role of microscopic chaos in the dynamics and relaxation of many-body quantum systems has made several experimentally confirmed predictions about the systems of interacting nuclear spins in solids, focusing, in particular, on the shapes of spin echo responses measured by nuclear magnetic resonance (NMR). These predictions were based on the idea that the transverse nuclear spin decays evolve in a manner governed at long times by the slowest decaying eigenmode of the quantum system, analogous to a chaotic resonance in a classical system. The present paper extends the above investigations both theoretically and experimentally. On the theoretical side, the notion of chaotic eigenmodes is used to make predictions about the relationships between the long-time oscillation phase of the nuclear free induction decay (FID) and the amplitudes and phases of spin echoes. On the experimental side, the above predictions are tested for the nuclear spin decays of F-19 in CaF2 crystals and Xe-129 in frozen xenon. Good agreement between the theory and the experiment is found.Comment: 20 pages, 9 figures, significant new experimental content in comparison with version