Limits and Confidence Intervals in the Presence of Nuisance Parameters

We study the frequentist properties of confidence intervals computed by the method known to statisticians as the Profile Likelihood. It is seen that the coverage of these intervals is surprisingly good over a wide range of possible parameter values for important classes of problems, in particular whenever there are additional nuisance parameters with statistical or systematic errors. Programs are available for calculating these intervals.Comment: 6 figure

Anatomy of the Higgs fits: a first guide to statistical treatments of the theoretical uncertainties

The studies of the Higgs boson couplings based on the recent and upcoming LHC data open up a new window on physics beyond the Standard Model. In this paper, we propose a statistical guide to the consistent treatment of the theoretical uncertainties entering the Higgs rate fits. Both the Bayesian and frequentist approaches are systematically analysed in a unified formalism. We present analytical expressions for the marginal likelihoods, useful to implement simultaneously the experimental and theoretical uncertainties. We review the various origins of the theoretical errors (QCD, EFT, PDF, production mode contamination...). All these individual uncertainties are thoroughly combined with the help of moment-based considerations. The theoretical correlations among Higgs detection channels appear to affect the location and size of the best-fit regions in the space of Higgs couplings. We discuss the recurrent question of the shape of the prior distributions for the individual theoretical errors and find that a nearly Gaussian prior arises from the error combinations. We also develop the bias approach, which is an alternative to marginalisation providing more conservative results. The statistical framework to apply the bias principle is introduced and two realisations of the bias are proposed. Finally, depending on the statistical treatment, the Standard Model prediction for the Higgs signal strengths is found to lie within either the $68\%$ or $95\%$ confidence level region obtained from the latest analyses of the $7$ and $8$ TeV LHC datasets.Comment: 62 pages, 10 figure

On large-sample estimation and testing via quadratic inference functions for correlated data

Hansen (1982) proposed a class of "generalized method of moments" (GMMs) for estimating a vector of regression parameters from a set of score functions. Hansen established that, under certain regularity conditions, the estimator based on the GMMs is consistent, asymptotically normal and asymptotically efficient. In the generalized estimating equation framework, extending the principle of the GMMs to implicitly estimate the underlying correlation structure leads to a "quadratic inference function" (QIF) for the analysis of correlated data. The main objectives of this research are to (1) formulate an appropriate estimated covariance matrix for the set of extended score functions defining the inference functions; (2) develop a unified large-sample theoretical framework for the QIF; (3) derive a generalization of the QIF test statistic for a general linear hypothesis problem involving correlated data while establishing the asymptotic distribution of the test statistic under the null and local alternative hypotheses; (4) propose an iteratively reweighted generalized least squares algorithm for inference in the QIF framework; and (5) investigate the effect of basis matrices, defining the set of extended score functions, on the size and power of the QIF test through Monte Carlo simulated experiments.Comment: 32 pages, 2 figure

Variable Selection for Nonparametric Gaussian Process Priors: Models and Computational Strategies

This paper presents a unified treatment of Gaussian process models that extends to data from the exponential dispersion family and to survival data. Our specific interest is in the analysis of data sets with predictors that have an a priori unknown form of possibly nonlinear associations to the response. The modeling approach we describe incorporates Gaussian processes in a generalized linear model framework to obtain a class of nonparametric regression models where the covariance matrix depends on the predictors. We consider, in particular, continuous, categorical and count responses. We also look into models that account for survival outcomes. We explore alternative covariance formulations for the Gaussian process prior and demonstrate the flexibility of the construction. Next, we focus on the important problem of selecting variables from the set of possible predictors and describe a general framework that employs mixture priors. We compare alternative MCMC strategies for posterior inference and achieve a computationally efficient and practical approach. We demonstrate performances on simulated and benchmark data sets.Comment: Published in at http://dx.doi.org/10.1214/11-STS354 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org