Search

The paper is concerned with inference for a parameter of interest in models that share a common interpretation for that parameter but that may differ appreciably in other respects. We study the general structure of models under which the maximum likelihood estimator of the parameter of interest is consistent under arbitrary misspecification of the nuisance part of the model. A specialization of the general results to matched-comparison and two-groups problems gives a more explicit and easily checkable condition in terms of a new notion of symmetric parametrization, leading to a broadening and unification of existing results in those problems. The role of a generalized definition of parameter orthogonality is highlighted, as well as connections to Neyman orthogonality. The issues involved in obtaining inferential guarantees beyond consistency are briefly discussed.

Composite likelihood usually ignores dependencies among response components, while variational approximation to likelihood ignores dependencies among parameter components. We derive a Gaussian variational approximation to the composite log-likelihood function for Poisson and Gamma regression models with crossed random effects. We show consistency and asymptotic normality of the estimates derived from this approximation and support this theory with some simulation studies. The approach is computationally much faster than a Gaussian variational approximation to the full log-likelihood function.

Unexpectedly strong X-ray emission from extragalactic radio jets on kiloparsec scales has been one of the major discoveries of Chandra, the only X-ray observatory capable of sub-arcsecond-scale imaging. The origin of this X-ray emission, which appears as a second spectral component from that of the radio emission, has been debated for over two decades. The most commonly assumed mechanism is inverse Compton upscattering of the Cosmic Microwave Background (IC-CMB) by very low-energy electrons in a still highly relativistic jet. Under this mechanism, no variability in the X-ray emission is expected. Here we report the detection of X-ray variability in the large-scale jet population, using a novel statistical analysis of 53 jets with multiple Chandra observations. Taken as a population, we find that the distribution of p-values from a Poisson model is strongly inconsistent with steady emission, with a global p-value of 1.96e-4 under a Kolmogorov-Smirnov test against the expected Uniform (0,1) distribution. These results strongly imply that the dominant mechanism of X-ray production in kpc-scale jets is synchrotron emission by a second population of electrons reaching multi-TeV energies. X-ray variability on the time-scale of months to a few years implies extremely small emitting volumes much smaller than the cross-section of the jet., Comment: Published in Nature Astronomy 29 May 2023; Supplemental Information and Excel File included in 1st submission; Published article in most recent

We develop a procedure that transforms any asymptotically normal estimator into an asymptotically normal estimator whose distribution is robust to arbitrary data contamination. More generally, our procedure transforms any estimator whose asymptotic distribution has positive and continuous density at the origin into an asymptotically normal estimator whose distribution is robust to arbitrary contamination. In developing such a procedure we prove new general properties of componentwise and geometric quantiles in both finite and infinite dimensions., Comment: 48 pages; comments are welcomed

We examine the normal approximation of the modified likelihood root, an inferential tool from higher-order asymptotic theory, for the linear exponential and location-scale family. We show that the $r^\star$ statistic can be thought of as a location and scale adjustment to the likelihood root $r$ up to $O_p(n^{-3/2})$, and more generally $r^\star$ can be expressed as a polynomial in $r$. We also show the linearity of the modified likelihood root in the likelihood root for these two families., Comment: 16 pages, 3 figures

We examine the behaviour of the Laplace and saddlepoint approximations in the high-dimensional setting, where the dimension of the model is allowed to increase with the number of observations. Approximations to the joint density, the marginal posterior density and the conditional density are considered. Our results show that under the mildest assumptions on the model, the error of the joint density approximation is $O(p^4/n)$ if $p = o(n^{1/4})$ for the Laplace approximation and saddlepoint approximation, and $O(p^3/n)$ if $p = o(n^{1/3})$ under additional assumptions on the second derivative of the log-likelihood. Stronger results are obtained for the approximation to the marginal posterior density., Comment: 56 pages

This paper develops an approach to inference in a linear regression model when the number of potential explanatory variables is larger than the sample size. The approach treats each regression coefficient in turn as the interest parameter, the remaining coefficients being nuisance parameters, and seeks an optimal interest-respecting transformation, inducing sparsity on the relevant blocks of the notional Fisher information matrix. The induced sparsity is exploited through a marginal least squares analysis for each variable, as in a factorial experiment, thereby avoiding penalization. One parameterization of the problem is found to be particularly convenient, both computationally and mathematically. In particular, it permits an analytic solution to the optimal transformation problem, facilitating theoretical analysis and comparison to other work. In contrast to regularized regression such as the lasso and its extensions, neither adjustment for selection nor rescaling of the explanatory variables is needed, ensuring the physical interpretation of regression coefficients is retained. Recommended usage is within a broader set of inferential statements, so as to reflect uncertainty over the model as well as over the parameters. The considerations involved in extending the work to other regression models are briefly discussed., Comment: 31 pages, 4 figures, 7 tables

The likelihood function is central to both frequentist and Bayesian formulations of parametric statistical inference, and large-sample approximations to the sampling distributions of estimators and test statistics, and to posterior densities, are widely used in practice. Improved approximations have been widely studied and can provide highly accurate inferences when samples are small or there are many nuisance parameters. This article reviews improved approximations based on the tangent exponential model developed in a series of articles by D.~A.~S.~Fraser and co-workers, attempting to explain the theoretical basis of this model and to provide a guide to the associated literature, including a partially-annotated bibliography.

Directional tests to compare incomplete undirected graphs are developed in the general context of covariance selection for Gaussian graphical models. The exactness of the underlying saddlepoint approximation is proved for chordal graphs and leads to exact control of the size of the tests, given that the only approximation error involved is due to the numerical calculation of two scalar integrals. Although exactness is not guaranteed for non-chordal graphs, the ability of the saddlepoint approximation to control the relative error leads the directional test to overperform its competitors even in these cases. The accuracy of our proposal is verified by simulation experiments under challenging scenarios, where inference via standard asymptotic approximations to the likelihood ratio test and some of its higher-order modifications fails. The directional approach is used to illustrate the assessment of Markovian dependencies in a dataset from a veterinary trial on cattle. A second example with microarray data shows how to select the graph structure related to genetic anomalies due to acute lymphocytic leukemia.

Directional inference for vector parameters based on higher order approximations in likelihood inference has recently been developed in the literature. Here we explore examples of directional inference where the calculations can be simplified. It is found that in several classical situations the directional test is equivalent to the usual $F$-test. These situations include testing the ratio of group variances and rates from normal and exponential distributions, linear regression with a linear constraint and Hotelling's $T^2$-test., Comment: 9 pages, 14 pages of supporting information

Does the asymptotic variance of the maximum composite likelihood estimator of a parameter of interest always decrease when the nuisance parameters are known? Will a composite likelihood necessarily become more efficient by incorporating addi- tional independent component likelihoods, or by using component likelihoods with higher dimension? In this note we show through illustrative examples that the an- swer to both questions is no, and indeed the opposite direction might be observed. The role of information bias is highlighted to understand the occurrence of these paradoxical phenomenon.

The need for new methods to deal with big data is a common theme in most scientific fields, although its definition tends to vary with the context. Statistical ideas are an essential part of this, and as a partial response, a thematic program on statistical inference, learning, and models in big data was held in 2015 in Canada, under the general direction of the Canadian Statistical Sciences Institute, with major funding from, and most activities located at, the Fields Institute for Research in Mathematical Sciences. This paper gives an overview of the topics covered, describing challenges and strategies that seem common to many different areas of application, and including some examples of applications to make these challenges and strategies more concrete., Comment: Thematic Program on Statistical Inference, Learning, and Models for Big Data, Fields Institute; 23 pages, 2 figures

I review the classical theory of likelihood based inference and consider how it is being extended and developed for use in complex models and sampling schemes., Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP03 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

The dominant theoretical frameworks (structural, human resources, political, or cultural) espoused by elementary school principals are studied; and the platforms are compared with data previously collected on leadership behaviors and time usage. Data for 5 principals and 151 teachers in 5 elementary schools in Philadelphia (Pennsylvania) on the Principal Instructional Management Rating Scale (PIMRS) are presented. The PIMRS is used to determine the teachers' perceptions of their principals' instructional management behaviors and perceptions of the principals. Data for principals on time use surveys are included. The elementary schools have student enrollments of between 382 and 816 students with minority populations from 16 percent to 100 percent. The espoused theoretical framework is determined using the principals' responses to the Leadership Orientations Instrument. The results show that one principal uses a structural frame, three principals use a human resources frame, and one principal uses a combination of structural, human resources, political, and symbolic frames. The principal who espoused the greatest number of theoretical frames has the most relevant and recent training. All the principals have internalized the norm of high expectations for students and teachers, and use this concept to drive their own leadership behaviors, as indicated by the teachers. In these urban schools, use of the political frame is surprisingly low. Five tables of statistical data and a 13-item list of references are included. (SLD)