Uniform Asymptotic Expansions of Certain Integrals

Handelsman, Richard A.

doi:10.1137/0115125

Cited by 420 publications

(734 citation statements)

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“…The results we obtain can also be considered as certain extensions of the Bayesian nonparametric estimations of medians presented in Doss (1985a, b) and the EL estimation proposed in Chen & Hall (1993). Note that proof schemes used in this article can be of independent interest to investigators dealing with applications of adapted Laplace methods (e.g., Bleinstein & Handelsman, 1975, Davison, 1986, Kass & Raftery, 1995) to handle step functions and EL functions. These proof schemes allow us to obtain asymptotic results similar to those of parametric BFs.…”

Section: Introductionmentioning

confidence: 67%

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Bayesian empirical likelihood methods for quantile comparisons

Vexler

Lazar

2017

Journal of the Korean Statistical Society

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Bayes factors, practical tools of applied statistics, have been dealt with extensively in the literature in the context of hypothesis testing. The Bayes factor based on parametric likelihoods can be considered both as a pure Bayesian approach as well as a standard technique to compute p-values for hypothesis testing. We employ empirical likelihood methodology to modify Bayes factor type procedures for the nonparametric setting. The paper establishes asymptotic approximations to the proposed procedures. These approximations are shown to be similar to those of the classical parametric Bayes factor approach. The proposed approach is applied towards developing testing methods involving quantiles, which are commonly used to characterize distributions. We present and evaluate one and two sample distribution free Bayes factor type methods for testing quantiles based on indicators and smooth kernel functions. An extensive Monte Carlo study and real data examples show that the developed procedures have excellent operating characteristics for one-sample and two-sample data analysis.

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Section: Introductionmentioning

confidence: 67%

“…The standard method to approximate integrals such as Ω is based on the Laplace technique (e.g., Bleinstein & Handelsman, 1975, Davison, 1986). This method requires log ℒ e ( q ) − log ( n − n ) to be continuous and twice differentiable.…”

Section: Methodsmentioning

confidence: 99%

Bayesian empirical likelihood methods for quantile comparisons

Vexler

Lazar

2017

Journal of the Korean Statistical Society

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“…We have the first-order perturbation results for the pressurerelease boundary condition (10) and the rigid boundary condition (11) The scattering function for the fluid-fluid case involves more nomenclature. To write the solution, let the fluid wavenumber in the upper halfspace be , and let the fluid wavenumber in the lower halfspace be .…”

Section: A Scattering Functionsmentioning

confidence: 99%

“…And, because the scattered field is spherically spreading in the far field, the double integral over the wavenumber plane shown in (8) can be understood more clearly by the change of variables and , where and . The integrals over and can then be approximated by the argument of stationary phase [10].…”

Section: A Scattering Functionsmentioning

confidence: 99%

Pseudorandom realization of transient acoustic waves scattered from randomly rough patches

Ricks

1997

IEEE J. Oceanic Eng.

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A simple numerical technique is developed for generating pseudorandom realizations of three-dimensional (3-D) transient acoustic waves that are scattered from two-dimensional (2-D) patches of randomly rough surfaces. The rough surface height of a patch is represented numerically in the 2-D horizontal wavenumber plane by choosing a scheme for interpolation between pseudorandom complex coefficients. Using this approach, the realizations of the patches can be generated from experimentally measured roughness power spectra, and phase information is generated in the frequency domain that leads to time spreads in the time domain. The acoustic scattering is modeled here with first-order perturbation theory. The boundary conditions considered here are pressure-release, rigid, and fluid-fluid. Three different spatial windows are considered for defining the patches. In the time domain, the time spreads of the scattered waveforms agree with predictions. In the frequency domain, the phase is seen as a random walk. The solutions developed here can be used with normal mode propagation models or ray propagation models.

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“…The methods of Chester, Friedman and Ursell have in turn been extended by Bleistein to treat a coalescing saddle point and branch point [4], and many coalescing saddle points and branch points [5]. Handelsman and Bleistein have also considered integrals where b(t, 1) 0 [6], double integrals with a stationary point [7], and integrals with stationary points which approach infinity [8]. A great variety of special functions appear in place of the Airy function, some of which apparently have not been studied before.…”

mentioning

confidence: 99%

Uniform Asymptotic Expansions for Wave Propagation and Diffracton Problems

Ludwig¹

1970

SIAM Rev.

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High frequency asymptotic methods lead to simplification of wave propagation problems because of a principle of localization" the high frequency behavior of solutions at a given point depends only upon local properties of the medium, boundaries and ray trajectories. This idea leads at once to the representation of solutions of complicated problems in terms of solutions of simpler, canonical problems. This idea has been pursued vigorously in recent years, and the following is an account of some of the work along these lines.In order to illustrate the basic ideas, we consider the asymptotic behavior of the Bessel function for large order and argument, i.e., we consider Jk(kr) for large k. The results may be interpreted as (i) asymptotic expansion of an integral, (ii) asymptotic solution of an ordinary differential equation, or (iii) asymptotic solution of a partial differential equation (since e-ikOJk(kr is a solution of Au + kZu 0). Here the emphasis will be on the third aspect, but there has been a continuing interaction between all three. 1. Uniform asymptotic expansion of integrals. With a suitable choice of contour, we have 1 fdk,(t,,. dr, (1.1) Jk(kr) where (1.2) (t, r) r sin t. For k >> 1 and r 1, the saddle point method is applicable (see Erd61yi [1): the saddle points i satisfy (1.3) c3t (' r) r cos 1 0. If r > 1, the saddle points are real and distinct, and we obtain the Debye approximations cos kx// 1-kcos --n/4 (1.4) Jk(kr) nkx//r2_ r If r < 1, the saddle points are complex conjugates, but the contour may be deformed to pick up only one contribution:(1.5) J(kr) 1 1/2 e -kcsh-(1/r)+k4/1 2kv/1 r 2

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Uniform Asymptotic Expansions of Certain Integrals

Cited by 420 publications

References 5 publications

Bayesian empirical likelihood methods for quantile comparisons

Bayesian empirical likelihood methods for quantile comparisons

Pseudorandom realization of transient acoustic waves scattered from randomly rough patches

Uniform Asymptotic Expansions for Wave Propagation and Diffracton Problems

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