Higher Order Asymptotics Research Articles

Amendments and Corrections: Higher-Order Asymptotics and the Likelihood Principle: One-Parameter Models

Amendments and Corrections: Higher-Order Asymptotics and the Likelihood Principle: One-Parameter Models

Comparing the performance of split-step algorithms for coastal acoustics problems

The split-step Fourier algorithm [R. H. Hardin and F. D. Tappert, SIAM Rev. 15, 423 (1973); S. M. Flatté and F. D. Tappert, J. Acoust. Soc. Am. 58, 1151–1159 (1975)] has been an important technique for solving range-dependent ocean acoustics problems for more than two decades. Since this algorithm is restricted to leading-order asymptotics, parabolic equation solutions based on finite-difference algorithms have been developed for handling effects that require higher-order asymptotics, such as wide propagation angles, extensive bottom interaction, and elastic and poroelastic waves. The split-step Fourier algorithm has remained in widespread use despite the improved accuracy and capability of finite-difference solutions. The split-step Padé algorithm [M. D. Collins, J. Acoust. Soc. Am. 93, 1736–1742 (1993); M. D. Collins, J. Acoust. Soc. Am. 96, 382–385 (1994)], which includes higher-order asymptotics, is orders of magnitude faster than standard finite-difference solutions. Results will be presented for representative test problems to illustrate that the split-step Padé algorithm is also more efficient than the split-step Fourier solution for the coastal acoustics problems that are currently of interest.

Higher-order asymptotics and the likelihood principle: One-parameter models

Inferences based on higher-order asymptotics, in contrast to those based on standard first-order methods, approximate exact inferences closely enough to depend on the reference set. We use this feature to quantify the effect of the reference set on exact inferences. In particular, the difference in P-values for two reference sets can be at most O(n−½), and it is seen that one can often identify the coefficient of n−½. Although this is useful from a foundational viewpoint, the question arises of whether higher-order adjustments to first-order P-values reflect only the choice of reference set. It is indicated that for one-parameter problems this may largely be the case, but that when there are nuisance paramenters the situation is likely to be quite different. The nuisance parameter setting is not investigated thoroughly in this paper, but we believe that in that case the most important part of the higher-order corrections largely conforms to the likelihood principle.

Frequentist Validity of Posterior Quantiles in the Presence of a Nuisance Parameter: Higher Order Asymptotics

Frequentist Validity of Posterior Quantiles in the Presence of a Nuisance Parameter: Higher Order Asymptotics

Conditional likelihood and power: higher-order asymptotics

This paper considers, in the presence of a nuisance parameter, a very large class of tests that includes the conditional and the usual versions of the likelihood ratio (LR), Rao’s and Wald’s tests. Under contiguous alternatives and orthogonal parametrization, the power functions of the conditional and the usual versions of these tests have been compared and, in particular, it is seen that the power functions of the conditional versions, unlike those of the usual versions, are identical, up to the second-order, with the power functions of the corresponding tests with known nuisance parameter. An optimality property of the conditional LR test, in terms of second-order local maximinity, has been established. A test, optimal in the sense of third-order average power under contiguous alternatives, has been proposed. A weaker optimality property of Rao’s test, in terms of third-order average power, has also been indicated.

Higher order asymptotics in estimation for two-sided Weibull type distributions

Higher order asymptotics in estimation for two-sided Weibull type distributions

High-frequency multiple diffraction by a flat strip: Higher order asymptotics

By including correction terms in inverse powers of the wavenumber k , one may hope to extend the range of applicability of multiple edge diffracted geometrical theory of diffraction (GTD) to lower frequencies, and also to extend thereby the range of validity of the corresponding time domain solutions. The correction can be applied to each of the surface rays in the hierarchy that has been proposed by us recently as a model for multiple interaction between parallel edges separated by a plane surface segment on a two-dimensional perfectly conducting scatterer. The surface rays, which were found to explain the structure of the complex resonances in transient scattering, are excited for each interaction by equivalent line sources, dipole line sources, and their derivatives, with strengths determined from the asymptotic expansion of the edge diffracted field. This procedure is applied in detail to E - and H - polarized plane wave scattering by a perfectly conducting flat strip, up to quadruple diffraction, including consistent O(k^{-2}) corrections with respect to the dominant term. The procedure is applied also to generate corrected multiple diffracted individual surface ray fields, which lead to an improved equation for the complex resonances in the layer synthesized in the complex frequency plane by a surface ray of a particular order. Inclusion of the low frequency corrections reduces further the already small discrepancy between the ray optically calculated low frequency resonances and those computed numerically by the moment method.

Geometrical theory of higher-order asymptotics of test, interval estimator and conditional inference

Higher order asymptotic powers of one-sided and two-sided tests, both conditional and unconditional, are evaluated for a one-parameter curved exponential family of distributions by using differential-geometrical notions. Higher order asymptotic powers and the expected size of one-sided and two-sided interval estimators, both conditional and unconditional, are also obtained. The tests and interval estimators, which are third-order most powerful at any arbitrary specified one point are explicitly designed with the help of the approximate ancillary statistic. The characteristics of the conditional inference are elucidated, by proving the equivalence up to the third order of the conditional inference and likelihood ratio inference. Geometrical notions such as curvatures and angles play a fundamental role in the present theory.

Mathematical Statistics: Basic Ideas and Selected Topics.

(NOTE: Each chapter concludes with Problems and Complements, Notes, and References.) 1. Statistical Models, Goals, and Performance Criteria. Data, Models, Parameters, and Statistics. Bayesian Models. The Decision Theoretic Framework. Prediction. Sufficiency. Exponential Families. 2. Methods of Estimation. Basic Heuristics of Estimation. Minimum Contrast Estimates and Estimating Equations. Maximum Likelihood in Multiparameter Exponential Families. Algorithmic Issues. 3. Measures of Performance. Introduction. Bayes Procedures. Minimax Procedures. Unbiased Estimation and Risk Inequalities. Nondecision Theoretic Criteria. 4. Testing and Confidence Regions. Introduction. Choosing a Test Statistic: The Neyman-Pearson Lemma. Uniformly Most Powerful Tests and Monotone Likelihood Ratio Models. Confidence Bounds, Intervals and Regions. The Duality between Confidence Regions and Tests. Uniformly Most Accurate Confidence Bounds. Frequentist and Bayesian Formulations. Prediction Intervals. Likelihood Ratio Procedures. 5. Asymptotic Approximations. Introduction: The Meaning and Uses of Asymptotics. Consistency. First- and Higher-Order Asymptotics: The Delta Method with Applications. Asymptotic Theory in One Dimension. Asymptotic Behavior and Optimality of the Posterior Distribution. 6. Inference in the Multiparameter Case. Inference for Gaussian Linear Models. Asymptotic Estimation Theory in p Dimensions. Large Sample Tests and Confidence Regions. Large Sample Methods for Discrete Data. Generalized Linear Models. Robustness Properties and Semiparametric Models. Appendix A: A Review of Basic Probability Theory. The Basic Model. Elementary Properties of Probability Models. Discrete Probability Models. Conditional Probability and Independence. Compound Experiments. Bernoulli and Multinomial Trials, Sampling with and without Replacement. Probabilities on Euclidean Space. Random Variables and Vectors: Transformations. Independence of Random Variables and Vectors. The Expectation of a Random Variable. Moments. Moment and Cumulant Generating Functions. Some Classical Discrete and Continuous Distributions. Modes of Convergence of Random Variables and Limit Theorems. Further Limit Theorems and Inequalities. Poisson Process. Appendix B: Additional Topics in Probability and Analysis. Conditioning by a Random Variable or Vector. Distribution Theory for Transformations of Random Vectors. Distribution Theory for Samples from a Normal Population. The Bivariate Normal Distribution. Moments of Random Vectors and Matrices. The Multivariate Normal Distribution. Convergence for Random Vectors: Op and Op Notation. Multivariate Calculus. Convexity and Inequalities. Topics in Matrix Theory and Elementary Hilbert Space Theory. Appendix C: Tables. The Standard Normal Distribution. Auxiliary Table of the Standard Normal Distribution. t Distribution Critical Values. X 2 Distribution Critical Values. F Distribution Critical Values. Index.