Higher Order Asymptotics Research Articles

Bounded tilting estimation

The search for one-step alternatives to the generalized method of moment (GMM) has identified broad classes of potential estimators such as generalized empirical likelihoods (GEL), empirical cressie-read (ECR), exponentially tilted empirical likelihood (ETEL), and minimum discrepancy (MD) estimators. While empirical likelihood (EL) dominates other ECR estimators in terms of higher-order asymptotics, it lacks robustness to model misspecification. ETEL was shown to combine higher-order efficiency and robustness to misspecification but demands strong moment generating function existence conditions. We show, both theoretically and via simulations, how to achieve the same goal under weaker moment existence conditions within the class of MD estimators.

Signal-to-Noise Ratio Aware Minimaxity and Higher-Order Asymptotics

Signal-to-Noise Ratio Aware Minimaxity and Higher-Order Asymptotics

Higher-order asymptotic corrections and their application to the Gamma Variance Model

We present improved methods for calculating confidence intervals and p values in situations where standard asymptotic approaches fail due to small sample sizes. We apply these techniques to a specific class of statistical model that can incorporate uncertainties in parameters that themselves represent uncertainties (informally, “errors on errors”) called the Gamma Variance Model. This model contains fixed parameters, generically denoted by ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}, that represent the relative uncertainties in estimates of standard deviations of Gaussian distributed measurements. If the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} parameters are small, one can construct confidence intervals and p values using standard asymptotic methods. This is formally similar to the familiar situation of a large data sample, in which estimators for all adjustable parameters have Gaussian distributions. Here we address the important case where the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} parameters are not small and as a consequence the first-order asymptotic distributions do not represent a good approximation. We investigate improved test statistics based on the technology of higher-order asymptotics (modified likelihood root and Bartlett correction). The effective application of higher-order corrections removes an important computational barrier to the use of the Gamma Variance Model.

Quantum-Field Multiloop Calculations in Critical Dynamics

The quantum-field renormalization group method is one of the most efficient and powerful tools for studying critical and scaling phenomena in interacting many-particle systems. The multiloop Feynman diagrams underpin the specific implementation of the renormalization group program. In recent years, multiloop computation has had a significant breakthrough in both static and dynamic models of critical behavior. In the paper, we focus on the state-of-the-art computational techniques for critical dynamic diagrams and the results obtained with their help. The generic nature of the evaluated physical observables in a wide class of field models is manifested in the asymptotic character of perturbation expansions. Thus, the Borel resummation of series is required to process multiloop results. Such a procedure also enables one to take high-order contributions into consideration properly. The paper outlines the resummation framework in dynamic models and the circumstances in which it can be useful. An important resummation criterion is the properties of the higher-order asymptotics of the perturbation theory. In static theories, these properties are determined by the method of instanton analysis. A similar approach is applicable in critical dynamics models. We describe the calculation of these asymptotics in dynamical models and present the results of the corresponding resummation.

On the stability of traveling wave solutions to thin-film and long-wave models for film flows inside a tube

Traveling wave solutions are studied numerically and theoretically for models of viscous core–annular flows and falling film flows inside a tube. The models studied fall into one of two classes, referred to here as ‘thin-film’ or ‘long-wave’. One model of each type is studied for three problems: a falling viscous film lining the inside of a tube, and core–annular flow with either equal- or unequal-density fluids. In recent work, traveling wave solutions for some of these equations were found using a smoothing technique that removes a degeneracy and allows for continuation onto a periodic family of solutions from a Hopf bifurcation. This paper has three goals. First, the smoothing technique used in earlier studies is justified for these models using asymptotics. Second, this technique is used to find numerically families of traveling wave solutions not previously explored in detail, including some which have multiple turning points due to the interaction between gravity, viscous forces, surface tension, and pressure-driven flow. Third, the stability of these solutions is studied using asymptotics near the Hopf bifurcation point, and numerically far from this point. In particular, a simple theory using the constant solution at the Hopf bifurcation point produces estimates for the eigenvalues in good agreement with numerics, with the exception of the eigenvalues closest to zero; higher-order asymptotics are used to predict these eigenvalues. Far from the Hopf bifurcation point, the stabilizing role of increasing surface tension is quantified numerically for the thin-film models, while multiple changes in stability occur along families of solutions for some of the long-wave models.

Can Bayesian, confidence distribution and frequentist inference agree?

We discuss and characterise connections between frequentist, confidence distribution and objective Bayesian inference, when considering higher-order asymptotics, matching priors, and confidence distributions based on pivotal quantities. The focus is on testing precise or sharp null hypotheses on a scalar parameter of interest. Moreover, we illustrate that the application of these procedures requires little additional effort compared to the application of standard first-order theory. In this respect, using the R software, we indicate how to perform in practice the computation with three examples in the context of data from inter-laboratory studies, of the stress–strength reliability, and of a growth curve from dose–response data.

The second-order bias of quantile estimators

The finite sample theory using higher-order asymptotics provides better approximations of the bias for a class of estimators. Phillips (1991) demonstrated the higher-order asymptotic expansions for LAD estimators. Rilstone et al. (1996) provided the second-order bias results of conditional mean regression estimators. This paper develops new analytical results on the second-order bias of the conditional quantile regression estimators, which enables an improved bias correction and thus to obtain improved quantile estimation. In particular, we show that the second-order bias is larger towards the tails of the conditional density than near the median, and therefore the benefit of the second-order bias correction is greater when we are interested in the deeper tail quantiles, e.g., for the study of income distribution and financial risk management. The Monte Carlo simulation confirms the theory that the bias is larger at the tail quantiles, and the second-order bias correction improves the behavior of the quantile estimators.

Higher-Order Asymptotics and Its Application to Testing the Equality of the Examinee Ability Over Two Sets of Items.

In educational and psychological measurement, researchers and/or practitioners are often interested in examining whether the ability of an examinee is the same over two sets of items. Such problems can arise in measurement of change, detection of cheating on unproctored tests, erasure analysis, detection of item preknowledge, etc. Traditional frequentist approaches that are used in such problems include the Wald test, the likelihood ratio test, and the score test (e.g., Fischer, Appl Psychol Meas 27:3-26, 2003; Finkelman, Weiss, & Kim-Kang, Appl Psychol Meas 34:238-254, 2010; Glas & Dagohoy, Psychometrika 72:159-180, 2007; Guo & Drasgow, Int J Sel Assess 18:351-364, 2010; Klauer & Rettig, Br J Math Stat Psychol 43:193-206, 1990; Sinharay, J Educ Behav Stat 42:46-68, 2017). This paper shows that approaches based on higher-order asymptotics (e.g., Barndorff-Nielsen & Cox, Inference and asymptotics. Springer, London, 1994; Ghosh, Higher order asymptotics. Institute of Mathematical Statistics, Hayward, 1994) can also be used to test for the equality of the examinee ability over two sets of items. The modified signed likelihood ratio test (e.g., Barndorff-Nielsen, Biometrika 73:307-322, 1986) and the Lugannani-Rice approximation (Lugannani & Rice, Adv Appl Prob 12:475-490, 1980), both of which are based on higher-order asymptotics, are shown to provide some improvement over the traditional frequentist approaches in three simulations. Two real data examples are also provided.

Interview with Nancy Reid

SummaryNancy Reid was born in September 1952 in Niagara Falls, Canada. She graduated from the University of Waterloo with a Bachelor in Mathematics and a major in Statistics in 1974. She pursued her training in Statistics at the University of British Columbia (UBC) where she obtained a Master's in Applied Mathematics in 1976 and at Stanford University, where she graduated with a PhD in Statistics in 1979. After spending one year at Imperial College in London visiting Sir David Cox, she joined UBC as an Assistant Professor in the Department of Mathematics, where she had an important role in the creation of the Department of Statistics. In 1986, she moved to the University of Toronto, where she has been since then as a faculty in the Department of Statistics. Nancy has served as Chair of the Department between 1997 and 2002. Nancy's research in conditional inference, higher order asymptotics and composite likelihood has been influential in Statistics. Her outstanding contributions to Statistics were recognized nationally and internationally with many awards, including the President's Award of the Committee of Presidents of Statistical Societies (COPSS), Gold Medal awarded by the Statistical Society of Canada and Elected Foreign Associate of the National Academy of Sciences. She received the Doctor of Mathematics, Honoris Causa, University of Waterloo. Nancy served with distinction as Editor of the Canadian Journal of Statistics and President of the Statistical Society of Canada and President of the Institute of Mathematical Statistics. In 2014, she was appointed as Officer of the Order of Canada for her outstanding achievements, exemplary leadership and service to Canadians. The following conversation took place at the JSM 2016 in Chicago, on August 2 and 3, 2016.

Improving likelihood-based inference in control rate regression.

Control rate regression is a diffuse approach to account for heterogeneity among studies in meta-analysis by including information about the outcome risk of patients in the control condition. Correcting for the presence of measurement error affecting risk information in the treated and in the control group has been recognized as a necessary step to derive reliable inferential conclusions. Within this framework, the paper considers the problem of small sample size as an additional source of misleading inference about the slope of the control rate regression. Likelihood procedures relying on first-order approximations are shown to be substantially inaccurate, especially when dealing with increasing heterogeneity and correlated measurement errors. We suggest to address the problem by relying on higher-order asymptotics. In particular, we derive Skovgaard's statistic as an instrument to improve the accuracy of the approximation of the signed profile log-likelihood ratio statistic to the standard normal distribution. The proposal is shown to provide much more accurate results than standard likelihood solutions, with no appreciable computational effort. The advantages of Skovgaard's statistic in control rate regression are shown in a series of simulation experiments and illustrated in a real data example. R code for applying first- and second-order statistic for inference on the slope on the control rate regression is provided.

Escape rates for special flows and their higher order asymptotics

In this paper escape rates and local escape rates for special flows are studied. In a general context the first result is that the escape rate depends monotonically on the ceiling function and fulfils certain scaling, invariance and continuity properties. In the context of metric spaces local escape rates are considered. If the base transformation is ergodic and exhibits an exponential convergence in probability of ergodic sums, then the local escape rate with respect to the flow is just the local escape rate with respect to the base transformation, divided by the integral of the ceiling function. Also, a reformulation with respect to induced pressure is presented. Finally, under additional regularity conditions higher order asymptotics for the local escape rate are established.

Asymptotics of parabolic Green’s functions on lattices

For parabolic spatially discrete equations, we consider Green's functions, also known as heat kernels on lattices. We obtain their asymptotic expansions with respect to powers of time variable $t$ up to an arbitrary order and estimate the remainders uniformly on the whole lattice. The spatially discrete (difference) operators under consideration are finite-difference approximations of continuous strongly elliptic differential operators (with constant coefficients) of arbitrary even order in $\mathbb R^d$ with arbitrary $d\in\mathbb N$. This genericity, besides numerical and deterministic lattice-dynamics applications, allows one to obtain higher-order asymptotics of transition probability functions for continuous-time random walks on $\mathbb Z^d$ and other lattices.

Upper and lower bounds for the correlation function via inducing with general return times

For non-uniformly expanding maps inducing with a general return time to Gibbs Markov maps, we provide sufficient conditions for obtaining higher-order asymptotics for the correlation function in the infinite measure setting. Along the way, we show that these conditions are sufficient to recover previous results on sharp mixing rates in the finite measure setting for non-Markov maps, but for a larger class of observables. The results are illustrated by (finite and infinite measure-preserving) non-Markov interval maps with an indifferent fixed point.

Operator renewal theory for continuous time dynamical systems with finite and infinite measure

We develop operator renewal theory for flows and apply this to obtain results on mixing and rates of mixing for a large class of finite and infinite measure semiflows. Examples of systems covered by our results include suspensions over parabolic rational maps of the complex plane, and nonuniformly expanding semiflows with indifferent periodic orbits. In the finite measure case, the emphasis is on obtaining sharp rates of decorrelations, extending results of Gouezel and Sarig from the discrete time setting to continuous time. In the infinite measure case, the primary question is to prove results on mixing itself, extending our results in the discrete time setting. In some cases, we obtain also higher order asymptotics and rates of mixing.

Higher order asymptotics of solutions of a singular ODE

Higher order asymptotics of solutions of a singular ODE

Mixing for invertible dynamical systems with infinite measure

In a recent paper, Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math.189 (2012) 61–110] obtained results on mixing and mixing rates for a large class of noninvertible maps preserving an infinite ergodic invariant measure. Here, we are concerned with extending these results to the invertible setting. Mixing is established for a large class of infinite measure invertible maps. Assuming additional structure, in particular exponential contraction along stable manifolds, it is possible to obtain good results on mixing rates and higher order asymptotics.

Higher-order asymptotics for scoring rules

In this paper we discuss higher-order asymptotic expansions for proper scoring rules generalizing results for likelihood quantities, but meanwhile bring in the difficulty caused by the failure of the information identity. In particular, we derive higher-order approximations to the distribution of the scoring rule estimator, of the scoring rule ratio test statistic and, for a scalar parameter of interest, of the signed scoring rule root statistic. From these expansions, a modified signed scoring rule root statistic is proposed. Examples are given illustrating the accuracy of the modified signed scoring rule root statistic with respect to first-order methods.

Higher-order Bayesian Approximations for Pseudo-posterior Distributions

The theory of higher-order asymptotics provides accurate approximations to posterior distributions for a scalar parameter of interest, and to the corresponding tail area, for practical use in Bayesian analysis. The aim of this article is to extend these approximations to pseudo-posterior distributions, e.g., posterior distributions based on a pseudo-likelihood function and a suitable prior, which are proved to be particularly useful when the full likelihood is analytically or computationally infeasible. In particular, from a theoretical point of view, we derive the Laplace approximation for a pseudo-posterior distribution, and for the corresponding tail area, for a scalar parameter of interest, also in the presence of nuisance parameters. From a computational point of view, starting from these higher-order approximations, we discuss the higher-order tail area (HOTA) algorithm useful to approximate marginal posterior distributions, and related quantities. Compared to standard Markov chain Monte Carlo methods, the main advantage of the HOTA algorithm is that it gives independent samples at a negligible computational cost. The relevant computations are illustrated by two examples.

The Spectrum of a Family of Fourth-Order Nonlinear Diffusions Near the Global Attractor

The thin film and quantum drift diffusion equations belong to a fourth-order family of evolution equations proposed in [21] to be analogous to the (second-order) porous medium family. They are 2-Wasserstein (=d 2) gradient flows of the generalized Fisher information I(v) just as the porous medium family was shown to be the d 2 gradient flow of the generalized entropy E(v) by Otto [41]. The identity aI(v) = bE(v) + |∇ d 2 E(v)|2/2 implies a Hess d 2 I(v *) = Hess d 2 E(v *)(b + Hess d 2 E(v *)) formally, when the equation is rescaled and linearized around the resulting self-similar critical profile v *. We couple this relation with the diagonalization of Hess d 2 E(v *) for the porous medium flow computed in [46]. This yields information about the leading- and higher-order asymptotics of the equation on R N which—outside of special cases—was inaccessible previously.

Higher-order accuracy of multiscale-double bootstrap for testing regions

We consider hypothesis testing for the null hypothesis being represented as an arbitrary-shaped region in the parameter space. We compute an approximate p -value by counting how many times the null hypothesis holds in bootstrap replicates. This frequency, known as bootstrap probability, is widely used in evolutionary biology, but often reported as biased in the literature. Based on the asymptotic theory of bootstrap confidence intervals, there have been some new attempts for adjusting the bias via bootstrap probability without direct access to the parameter value. One such an attempt is the double bootstrap which adjusts the bias by bootstrapping the bootstrap probability. Another new attempt is the multiscale bootstrap which is similar to the m -out-of- n bootstrap but very unusually extrapolating the bootstrap probability to m = − n . In this paper, we employ these two attempts at the same time, and call the new procedure as multiscale-double bootstrap. By focusing on the multivariate normal model, we investigate higher-order asymptotics up to fourth-order accuracy. Geometry of the region plays important roles in the asymptotic theory. It was known in the literature that the curvature of the boundary surface of the region determines the bias of bootstrap probability. We found out that the “curvature of curvature” determines the remaining bias of double bootstrap. The multiscale bootstrap removes these biases. The multiscale-double bootstrap is fourth order accurate with coverage probability erring only O ( n − 2 ) , and it is robust against computational error of parameter estimation used for generating bootstrap replicates from the null distribution.