Second-order Asymptotic Expansion Research Articles

Asymptotic and measured large frequency separations

With the space-borne missions CoRoT and Kepler, a large amount of asteroseismic data is now available. So-called global oscillation parameters are inferred to characterize the large sets of stars, to perform ensemble asteroseismology, and to derive scaling relations. The mean large separation is such a key parameter. It is therefore crucial to measure it with the highest accuracy. As the conditions of measurement of the large separation do not coincide with its theoretical definition, we revisit the asymptotic expressions used for analysing the observed oscillation spectra. Then, we examine the consequence of the difference between the observed and asymptotic values of the mean large separation. The analysis is focused on radial modes. We use series of radial-mode frequencies to compare the asymptotic and observational values of the large separation. We propose a simple formulation to correct the observed value of the large separation and then derive its asymptotic counterpart. We prove that, apart from glitches due to stellar structure discontinuities, the asymptotic expansion is valid from main-sequence stars to red giants. Our model shows that the asymptotic offset is close to 1/4, as in the theoretical development. High-quality solar-like oscillation spectra derived from precise photometric measurements are definitely better described with the second-order asymptotic expansion. The second-order term is responsible for the curvature observed in the \'echelle diagrams used for analysing the oscillation spectra and this curvature is responsible for the difference between the observed and asymptotic values of the large separation. Taking it into account yields a revision of the scaling relations providing more accurate asteroseismic estimates of the stellar mass and radius.

Trace Approximations of Products of Truncated Toeplitz Operators

The paper establishes error orders for integral limit approximations to the traces of products of truncated Toeplitz operators generated by integrable real symmetric functions defined on the real line. These approximations and the corresponding error bounds are of importance in the statistical analysis of continuous-time stationary processes (asymptotic distributions and large deviations of Toeplitz-type quadratic functionals, estimation of the spectral parameters and functionals, etc.). An explicit second-order asymptotic expansion is found for the trace of a product of two truncated Toeplitz operators generated by the spectral densities of continuous-time stationary fractional Riesz–Bessel motions. The order of magnitude of the second term in this expansion is shown to depend on the long-memory parameters of the processes. Also, it is shown that the pole in the first-order approximation is removed by the second-order term, which provides a substantially improved approximation to the original functional.

OPTIMAL DESIGN OF LOW-CONTRAST TWO-PHASE STRUCTURES FOR THE WAVE EQUATION

This paper is concerned with the following optimal design problem: find the distribution of two phases in a given domain that minimizes an objective function evaluated through the solution of a wave equation. This type of optimization problem is known to be ill-posed in the sense that it generically does not admit a minimizer among classical admissible designs. Its relaxation could be found, in principle, through homogenization theory but, unfortunately, it is not always explicit, in particular for objective functions depending on the solution gradient. To circumvent this difficulty, we make the simplifying assumption that the two phases have a low constrast. Then, a second-order asymptotic expansion with respect to the small amplitude of the phase coefficients yields a simplified optimal design problem which is amenable to relaxation by means of H-measures. We prove a general existence theorem in a larger class of composite materials and propose a numerical algorithm to compute minimizers in this context. As in the case of an elliptic state equation, the optimal composites are shown to be rank-one laminates. However, the proof that relaxation and small-amplitude limit commute is more delicate than in the elliptic case.

Asymptotic expansion for nonparametric M-estimator in a nonlinear regression model with long-memory errors

We consider asymptotic expansion of the nonparametric M-estimator in a fixed-design nonlinear regression model when the errors are generated by long-memory linear processes. Under mild conditions, we show that the nonparametric M-estimator is first-order equivalent to the Nadaraya–Watson (NW) estimator, which implies that the nonparametric M-estimator has the same asymptotic distribution as that of the NW estimator. Furthermore, we study the second-order asymptotic expansion of the nonparametric M-estimator and show that the difference between the nonparametric M-estimator and the NW estimator has a limiting distribution after suitable standardization. The nature of the limiting distribution depends on the range of long-memory parameter α . We also compare the finite sample behavior of the two estimators through a numerical example when the errors are long-memory.

Asymptotic expansion of solutions to the drift-diffusion equation with large initial data

We consider the large-time behavior of the solution to the initial value problem for the Nernst–Planck type drift-diffusion equation in whole spaces. In the L p -framework, the global existence and the decay of the solution were shown. Moreover, the second-order asymptotic expansion of the solution as t → ∞ was derived. We also deduce the higher-order asymptotic expansion of the solution. Especially, we discuss the contrast between the odd-dimensional case and the even-dimensional case.

A Bayesian approach to the estimation of maps between Riemannian manifolds. II: Examples

Let gJ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space E s, and let γ be a smooth map of gJ into a Riemannian manifold Λ. An unknown state θ ∈ gJ is observed via X = θ + ɛξ, where ɛ > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on gJ and smooth estimators g(X) of the map γ we have derived a second-order asymptotic expansion for the related Bayesian risk [3]. In this paper, we apply this technique to a variety of examples.The second part examines the first-order conditions for equality-constrained regression problems. The geometric tools that are utilized in [3] are naturally applicable to these regression problems.

First- and second-order topological sensitivity analysis for inclusions

The topological derivative provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal perturbations. Therefore, as a natural extension of this concept, we can consider higher-order terms in the expansion. In particular, the next one we recognize as the second-order topological derivative, which allows us to deal with perturbations of finite sizes. This term depends explicitly on higher-order gradients of the solution associated to the non-perturbed problem and also implicitly through the solution of an auxiliary variational problem. In this article, we calculate the explicit as well as implicit terms of the second-order topological asymptotic expansion for the total potential energy associated to the Laplace equation in the two-dimensional domain. The domain perturbation is done by the insertion of a small inclusion with a thermal conductivity coefficent value different from the bulk material. Finally, we present some numerical experiments showing the influence of the second-order term in the topological asymptotic expansion for several values of the thermal conductivity coefficent of the inclusion.

On the convergence rate of sequential fixed-width confidence intervals for normal parameters

We consider the convergence rate of sequential fixed-width confidence intervals for θ = a μ + b σ under a normal N ( μ , σ 2 ) model with μ and σ 2 both unknown. We use the fully sequential procedure proposed by [Takada, Y., 1997. Fixed-width confidence intervals for a function of normal parameters. Sequential Anal. 16, 107–117] to construct sequential confidence intervals for θ and investigate the convergence rate of the coverage probability as the width of the confidence interval approaches zero. We also derive a second-order asymptotic expansion of the average sample size.

A Bayesian approach to the estimation of maps between Riemannian manifolds

Let \Theta be a smooth compact oriented manifold without boundary, embedded in a euclidean space and let \gamma be a smooth map \Theta into a riemannian manifold \Lambda. An unknown state \theta \in \Theta is observed via X=\theta+\epsilon \xi where \epsilon>0 is a small parameter and \xi is a white Gaussian noise. For a given smooth prior on \Theta and smooth estimator g of the map \gamma we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces \Theta and \Lambda, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of \gamma is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.

Semi-parametric second-order efficient estimation of the period of a signal

This paper is concerned with the estimation of the period of an unknown periodic function in Gaussian white noise. A class of estimators of the period is constructed by means of a penalized maximum likelihood method. A second-order asymptotic expansion of the risk of these estimators is obtained. Moreover, the minimax problem for the second-order term is studied and an estimator of the preceding class is shown to be second order efficient.

On the theoretical comparison of low-bias steady-state estimators

The time-average estimator is typically biased in the context of steady-state simulation, and its bias is of order 1/ t , where t represents simulated time. Several “low-bias” estimators have been developed that have a lower order bias, and, to first-order, the same variance of the time-average. We argue that this kind of first-order comparison is insufficient, and that a second-order asymptotic expansion of the mean square error (MSE) of the estimators is needed. We provide such an expansion for the time-average estimator in both the Markov and regenerative settings. Additionally, we provide a full bias expansion and a second-order MSE expansion for the Meketon--Heidelberger low-bias estimator, and show that its MSE can be asymptotically higher or lower than that of the time-average depending on the problem. The situation is different in the context of parallel steady-state simulation, where a reduction in bias that leaves the first-order variance unaffected is arguably an improvement in performance.

Lower-Bound Estimates for Poisson-Type Approximations

Let S = w 1 S 1 + w 2 S 2 + ⋯ + w N S N . Here S j is a sum of identically distributed random variables with weight w j > 0. We consider the cases where S j is a sum of independent random variables, the sum of independent lattice variables, or has the Markov binomial distribution. Apart from the general case, we investigate the case of symmetric random variables. Distribution of S is approximated by a compound Poisson distribution, by a second-order asymptotic expansion, and by a signed exponential measure. Lower bounds for the accuracy of approximations in uniform metric are established.

The accuracy of stochastic perturbation solutions to subsurface transport problems

Perturbation methods are of interest to hydrologists because they provide a way to incorporate upscaling and accuracy assessment capabilities into practical groundwater models. In particular, these methods may be used to obtain approximate solutions for the ensemble moments of solute concentration and for related effective properties in randomly heterogeneous porous media. Although the many perturbation methods proposed in the groundwater literature seem to be rather different they are actually closely related. For example, we show here through a systematic analysis that the second-order asymptotic expansion approach [Flow and Transport in Porous Formations, Springer-Verlag, Berlin, 1989], random Green’s function methods [Dispersion in Heterogeneous Geological Formations, Kluwer Academic Publishers, the Netherlands, 2001], and Taylor series methods [Water Resour. Res. 32(1) (1996) 85; Water Resour. Res. 28(12) (1992) 3269; Water Resour. Res. 21(3) (1985) 359] all give identical expressions for the ensemble mean, for random fluctuations about this mean, and for related closure covariances. The Eulerian truncation approach [Water Resour. Res. 14(2) (1978) 263; Water Resour. Res. 19(1) (1983) 161; Phys. Fluids 31 (1988) 965; Water Resour. Res. 27(7) (1991) 1598] gives expressions similar to the other methods except that its closure covariances depend on the unknown ensemble mean rather than the known deterministic solution (the solution obtained when random velocities are replaced by their means). This reflects the fact that Eulerian truncation works with perturbations taken about the mean while the other methods work with perturbations taken about the deterministic solution. The ensemble mean accounts for the large-scale effects of small-scale variability while the deterministic solution does not. Eulerian truncation solutions can be more difficult to compute than other alternatives but they can also be more accurate. This is demonstrated for the case of Gaussian velocities with a general truncation analysis and with a specific numerical example. The results provided here suggest that Eulerian truncation is superior to alternative perturbation methods of comparable computational complexity.

Second-order asymptotic expansion for the risk in classification of curved exponential populations

This paper deals with the problem of classifying an observation into one of two curved exponential populations. The plug-in classification rule obtained by replacing unknown parameters with their bias-adjusted ML estimators in Bayes classification rule is used. The second-order asymptotic expansion with respect to the inverses of the training sample sizes for the expected regret risk is derived.

Second-order expansion for the expected regret risk in classification of one-parametric distributions

The plug-in rule is used for the classification of random observations into one of two regular one-parametric distributions. The maximum likelihood estimates of unknown parameters obtained from the stratified training sample are used. The second-order asymptotic expansion in terms of the inverses of the training sample sizes is derived for the expected regret risk. The closed-form expressions of the expansion coefficients are applicable for the performance evaluation of the proposed classification rule.

Predicting a future lifetime through Box-Cox transformation.

In predicting a future lifetime based on a sample of past lifetimes, the Box-Cox transformation method provides a simple and unified procedure that is shown in this article to meet or often outperform the corresponding frequentist solution in terms of coverage probability and average length of prediction intervals. Kullback-Leibler information and second-order asymptotic expansion are used to justify the Box-Cox procedure. Extensive Monte Carlo simulations are also performed to evaluate the small sample behavior of the procedure. Certain popular lifetime distributions, such as Weibull, inverse Gaussian and Birnbaum-Saunders are served as illustrative examples. One important advantage of the Box-Cox procedure lies in its easy extension to linear model predictions where the exact frequentist solutions are often not available.

The telegraph equation in charged particle transport

We present a new derivation of the telegraph equation which modifies its coefficients. First, an infinite order partial differential equation is obtained for the velocity space solid angle-averaged phase-space distribution of particles which underwent at least a few collisions. It is shown that, in the lowest order asymptotic expansion, this equation simplifies to the well-known diffusion equation. The second-order asymptotic expansion for isotropic small-angle scattering results in a modified telegraph equation with a signal propagation speed of v(5/11) exp 1/2 instead of the usual v/3 exp 1/2. Our derivation of a modified telegraph equation follows from an expansion of the Boltzmann equation in the relevant smallness parameters and not from a truncation of an eigenfunction expansion. This equation is consistent with causality. It is shown that, under steady state conditions in a convecting plasma, the telegraph equation may be regarded as a diffusion equation with a modified transport coefficient, which describes a combination of diffusion and cosmic-ray inertia.

Approximate Crack Opening Displacement Solutions for Long Circumferential Cracks in Pipes Subjected to Bending and Tension

A simple parabolic formula that describes the opening profile of a sufficiently long circumferential crack in a pipe subjected to bending and tension was found after a proper normalization of the second-order asymptotic expansion of the solutions of Sanders and Alabi [5-9]. Here the crack is sufficiently long in the sense that approximately the crack angle is larger than a small parameter which has the same order of magnitude as the ratio of the wall thickness to the pipe radius, and the pipe is sufficiently long enough not to consider the end effects. These linear elastic solutions can be superposed for combined bending and tension loading. The smile parabolic formula is shown to be a good approximation of the crack opening profile and can be used to evaluate the leak rate for crack length up to one-half of the circumference.

The Joint Distribution of Forecast Errors in the AR(1) Model

Second-order asymptotic expansion approximations to the joint distributions of dynamic forecast errors and of static forecast errors in the stationary Gaussian pure AR(1) model are derived. The approximation to the dynamic forecast errors distribution can be expressed as a multivariate normal distribution with modified mean vector and covariance matrix, thus generalizing the results of Phillips [12]. However, the approximation to the static forecast errors distribution includes skewness and kurtosis terms. Thus the class of multivariate normal distributions does not provide as good approximations (in terms of error convergence rates) to the distributions of the static forecast errors as to the distributions of the dynamic forecast errors. These results cast some doubt on the appropriateness of model validation procedures, such as Chow tests, which use the static forecast errors and implicitly assume that these have a distribution which is well approximated by a multivariate normal.

Easily determining which urns are ‘favorable’

The optimal sampling strategy for an urn, containing known numbers of plus and minus ones, can be simply described with the use of an empirically justified rule, based upon what appears to be a legitimate third-order asymptotic expansion of “the optimal stopping boundary” as the urn size goes to infinity performs exceedingly well. There is a known first-order asymptotic expansion due to Shepp. The reader is invited to try to justify a second-order asymptotic expansion of a type described by Chernoff and Petkau. The evidence presented in its support is very persuasive.