In Fisher's net : exact F-tests in semi-parametric models with exchangeable errors

We consider testing about the slope parameter β when Y - X β is assumed to be an exchangeable process conditionally on X. This framework encompasses the semi-parametric linear regression model. We show that the usual Fisher's procedure have non trivial exact rejection bound under the null hypothesis R β = ϒ. This bound derives from the Markov inequality and a close inspection of multivariate moments of self-normalized, self-centered, exchangeable processes. Improvement by higher order versions of the Markov inequality are also presented. The bounds do not require the existence of any moment, so they remain valid even if TCL do not apply. We generalize the framework to multivariate and order-1 auto-regressive models with exogenous variables

Background reionization history from omniscopes

The measurements of the 21-cm brightness temperature fluctuations from the neutral hydrogen at the Epoch of Reionization (EoR) should inaugurate the next generation of cosmological observables. In this respect, many works have concentrated on the disambiguation of the cosmological signals from the dominant reionization foregrounds. However, even after perfect foregrounds removal, our ignorance on the background reionization history can significantly affect the cosmological parameter estimation. In particular, the interdependence between the hydrogen ionized fraction, the baryon density and the optical depth to the redshift of observation induce nontrivial degeneracies between the cosmological parameters that have not been considered so far. Using a simple, but consistent reionization model, we revisit their expected constraints for a futuristic giant 21-cm omniscope by using for the first time Markov Chain Monte Carlo (MCMC) methods on multiredshift full sky simulated data. Our results agree well with the usual Fisher matrix analysis on the three-dimensional flat sky power spectrum but only when the above-mentioned degeneracies are kept under control. In the opposite situation, Fisher results can be inaccurate. We show that these conditions can be fulfilled by combining cosmic microwave background measurements with multiple observation redshifts probing the beginning of EoR. This allows a precise reconstruction of the total optical depth, reionization duration and maximal spin temperature. Finally, we discuss the robustness of these results in presence of unresolved ionizing sources. Although most of the standard cosmological parameters remain weakly affected, we find a significant degradation of the background reionization parameter estimation in presence of nuisance ionizing sources.Comment: 22 pages, 18 figures, uses RevTex. References added, matches published versio

The Available Information for Invariant Tests of a Unit Root

This paper considers the information available to invariant unit root tests at and near the unit root. Since all invariant tests will be functions of the maximal invariant, the Fisher information in this statistic will be the available information. The main finding of the paper is that the available information for all tests invariant to a linear trend is zero at the unit root. This result applies for any sample size, over a variety of distributions and correlation structures and is robust to the inclusion of any other deterministic component. In addition, an explicit bound upon the power of all invariant unit root tests is shown to depend solely upon the information. This bound is illustrated via comparison with the local-to-unity power envelope and a brief simulation study illustrates the impact that the requirements of invariance have on power.

Higher order semiparametric frequentist inference with the profile sampler

We consider higher order frequentist inference for the parametric component of a semiparametric model based on sampling from the posterior profile distribution. The first order validity of this procedure established by Lee, Kosorok and Fine in [J. American Statist. Assoc. 100 (2005) 960--969] is extended to second-order validity in the setting where the infinite-dimensional nuisance parameter achieves the parametric rate. Specifically, we obtain higher order estimates of the maximum profile likelihood estimator and of the efficient Fisher information. Moreover, we prove that an exact frequentist confidence interval for the parametric component at level $\alpha$ can be estimated by the $\alpha$-level credible set from the profile sampler with an error of order $O_P(n^{-1})$. Simulation studies are used to assess second-order asymptotic validity of the profile sampler. As far as we are aware, these are the first higher order accuracy results for semiparametric frequentist inference.Comment: Published in at http://dx.doi.org/10.1214/07-AOS523 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Semiparametric posterior limits

We review the Bayesian theory of semiparametric inference following Bickel and Kleijn (2012) and Kleijn and Knapik (2013). After an overview of efficiency in parametric and semiparametric estimation problems, we consider the Bernstein-von Mises theorem (see, e.g., Le Cam and Yang (1990)) and generalize it to (LAN) regular and (LAE) irregular semiparametric estimation problems. We formulate a version of the semiparametric Bernstein-von Mises theorem that does not depend on least-favourable submodels, thus bypassing the most restrictive condition in the presentation of Bickel and Kleijn (2012). The results are applied to the (regular) estimation of the linear coefficient in partial linear regression (with a Gaussian nuisance prior) and of the kernel bandwidth in a model of normal location mixtures (with a Dirichlet nuisance prior), as well as the (irregular) estimation of the boundary of the support of a monotone family of densities (with a Gaussian nuisance prior).Comment: 47 pp., 1 figure, submitted for publication. arXiv admin note: substantial text overlap with arXiv:1007.017

On Lower Bounds for Non Standard Deterministic Estimation

We consider deterministic parameter estimation and the situation where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on random variables as well. Unfortunately, in the general case, this marginalization is mathematically intractable, which prevents from using the known standard deterministic lower bounds (LBs) on the mean squared error (MSE). Actually the general case can be tackled by embedding the initial observation space in a hybrid one where any standard LB can be transformed into a modified one fitted to nonstandard deterministic estimation, at the expense of tightness however. Furthermore, these modified LBs (MLBs) appears to include the submatrix of hybrid LBs which is an LB for the deterministic parameters. Moreover, since in the nonstandard estimation, maximum likelihood estimators (MLEs) can be no longer derived, suboptimal nonstandard MLEs (NSMLEs) are proposed as being a substitute. We show that any standard LB on the MSE of MLEs has a nonstandard version lower bounding the MSE of NSMLEs. We provide an analysis of the relative performance of the NSMLEs, as well as a comparison with the MLBs for a large class of estimation problems. Last, the general approach introduced is exemplified, among other things, with a new look at the well-known Gaussian complex observation models