Asymptotic expansion of the null distribution of test statistic for linear hypothesis in nonnormal linear model

This paper is concerned with the null distribution of test statistic T for testing a linear hypothesis in a linear model without assuming normal errors. The test statistic includes typical ANOVA test statistics. It is known that the null distribution of T converges to χ² when the sample size n is large under an adequate condition of the design matrix. We extend this result by obtaining an asymptotic expansion under general condition. Next, asymptotic expansions of one- and two-way test statistics are obtained by using this general one. Numerical accuracies are studied for some approximations of percent points and actual test sizes of T for two-way ANOVA test case based on the limiting distribution and an asymptotic expansion.     

On the distributions of some test criteria for a covariance matrix under local alternatives and bootstrap approximations

The asymptotic distribution of some test criteria for a covariance matrix are derived under local alternatives. Except for the existence of some higher moments, no assumption as to the form of the distribution function is made. As an illustration, a case of t distribution included normal model is considered and the power of the likelihood ratio test and Nagao's test for sphericity, as described in Srivastava and Khatri and Anderson, is computed. Also, the power is computed using the bootstrap method. In the case of t distribution, the bootstrap approximation does not appear to be as good as the one obtained by the asymptotic expansion method.     

Third-order asymptotic expansion of <Emphasis Type="Italic">M</Emphasis>-estimators for diffusion processes

For an unknown parameter in the drift function of a diffusion process, we consider an M-estimator based on continuously observed data, and obtain its distributional asymptotic expansion up to the third order. Our setting covers the misspecified cases. To represent the coefficients in the asymptotic expansion, we derive some formulas for asymptotic cumulants of stochastic integrals, which are widely applicable to many other problems. Furthermore, asymptotic properties of cumulants of mixing processes will be also studied in a general setting.     

Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe

The nonsteady Navier–Stokes equations are considered in a thin infinite pipe with the small diameter ? in the case of the Reynolds number of order ?. The time-dependent flow rate is a given function. The complete asymptotic expansion is constructed and justified. The error estimate of order O(? ^J ) for the difference of the exact solution and the J-th asymptotic approximation is proved for any real J.     

Heat Content Asymptotics with Inhomogeneous Neumann and Dirichlet Boundary Conditions

Let M be a compact manifold with smooth boundary. We establish the existence of an asymptotic expansion for the heat content asymptotics of M with inhomogeneous Neumann and Dirichlet boundary conditions. We prove all the coefficients are locally determined and determine the first several terms in the asymptotic expansion.     

Large-scale structures as gradient lines: The case of the trkal flow

Based on expansion terms of the Beltrami-flow type, we use multiscale methods to effectively construct an asymptotic expansion at large Reynolds numbers R for the long-wavelength perturbation of the nonstationary anisotropic helical solution of the force-free Navier—Stokes equation (the Trkal solution). We prove that the systematic asymptotic procedure can be implemented only in the case where the scaling parameter is R ^1/2. Projections of quasistationary large-scale streamlines on a plane orthogonal to the anisotropy direction turn out to be the gradient lines of the energy density determined by the initial conditions for two modulated anisotropic Beltrami flows (modulated as a result of scaling) with the same eigenvalues of the curl operator. The three-dimensional streamlines and the curl lines, not coinciding, fill invariant vorticity tubes inside which the velocity and vorticity vectors are collinear up to terms of the order of 1/R.     

The likelihood ratio criterion and the asymptotic expansion of its distribution

Summary Asymptotic expansion of the distribution of the likelihood ratio criterion (LRC) for testing a composite hypothesis is derived under null hypothesis and a correction factor ρ which makes the term of order 1/n in the asymptotic expansion of the distribution of it vanish is obtained. The problem is extended to the case of a general composite hypothesis and of Pitman's local alternatives. The asymptotic distribution of LRC for a simple hypothesis is studied under a fixed alternative. The Institute of Statistical Mathematics     

On the Asymptotic Expansion of Hadamard Products

We consider functions f and g which are holomoxphic on closed sectors in C where they admit an asymptotic representation at ∞ in the form of power series in z^-1. We give a simple geometrical condition under which the Hadamard product f*g of f and g porsesses again an asymp totic expansion at ∞. It turns out that the asymptotic expansion of f*g is essentially the formal Hsdamd product of the asymptotic expansions of f and g. Our result yields a slight generaliestion of a well known theorem of W. B. Ford.     

Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points

Summary. A general formula is proved, which relates the equiaffine inner parallel curves of a plane convex body and the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points. This formula is applied to improve some well-known results in geometric probability. For example, an estimate, which was established for a special case by L. C. G. Rogers, is obtained with the best possible bound, and an asymptotic formula due to A. Rényi and R.␣Sulanke is extended to an asymptotic expansion. Received: 21 May 1996     

An acceleration property of theE-algorithm for alternate sequences

We prove a convergence acceleration result by theE-algorithm for sequences whose error has an asymptotic expansion on the scale of comparison for which a determinantal relation holds. This result is generalized to the vector case. Moreover we prove a result which contains an acceleration property for columns and diagonals of theE array. This result is applied to some alternating series.     

Asymptotic Integration of Parabolic Problems with Large High-Frequency Summands

We develop the averaging method theory for parabolic problems with rapidly oscillating summands some of which are large, i.e., proportional to the square root of the frequency of oscillations. In this case the corresponding averaged problems do not coincide in general with those obtained by the traditional averaging, i.e., by formally averaging the summands of the initial problem (since the principal term of the asymptotic expansion of a solution to the latter problem is not in general a solution to the so-obtained problem). In this article we consider the question of time periodic solutions to the first boundary value problem for a semilinear parabolic equation of an arbitrary order 2k whose nonlinear terms, including the large, depend on the derivatives of the unknown up to the order k-1. We construct the averaged problem and the formal asymptotic expansion of a solution. When the large summands depend on the unknown rather than its derivatives we justify the averaging method and the complete asymptotic expansion of a solution.Original Russian Text Copyright © 2005 Levenshtam V. B.The author was supported by the Russian Foundation for Basic Research (Grant 01-01-00678) and the Program “ Universities of Russia” (UR.04.01.029).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 805–821, July–August, 2005.     

An asymptotic expansion for the error in a linear map that reproduces polynomials of a certain order

Han's ‘multinode higher-order expansion’ in [H] is shown to be a special case of an asymptotic error expansion available for any bounded linear map on C([a..b]) that reproduces polynomials of a certain order. The key is the formula for the divided difference at a sequence containing just two distinct points.     

Malliavin calculus, geometric mixing, and expansion of diffusion functionals

Under geometric mixing condition, we presented asymptotic expansion of the distribution of an additive functional of a Markov or an ε-Markov process with finite autoregression including Markov type semimartingales and time series models with discrete time parameter. The emphasis is put on the use of the Malliavin calculus in place of the conditional type Cramér condition, whose verification is in most case not easy for continuous time processes without such an infinite dimensional approach. In the second part, by means of the perturbation method and the operational calculus, we proved the geometric mixing property for non-symmetric diffusion processes, and presented a sufficient condition which is easily checked in practice. Accordingly, we obtained asymptotic expansion of diffusion functionals and proved the validity of it under mild conditions, e.g., without the strong contractivity condition. Received: 7 September 1997 / Revised version: 17 March 1999     

The asymptotic behavior of solutions of the sine-gordon equation with singularities at zero

The problem of asymptotic analysis of radially symmetric solutions of the sine-Gordon equation reducible to the third Painlevé transcendent is posed. Solutions with singularities at the origin are studied. For finite values of the independent variable, an asymptotic expansion of such a solution is obtained; the leading term of this expansion is a modulated elliptic function. The corresponding modulation equation and phase shift are written out. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 329–342, March, 2000.     

A Variational Principle for the Ackerberg-O'Malley Resonance Problem

A new variational principle is proposed for determining the asymptotic expansion of the solution of the Ackerberg-O'Malley resonance problem [Stud. Appl. Math. 49:277–295 (1970)] to any order in ε. The method used yields new higher-order results not permitted by the technique of Grasman and Matkowsky [SIAM J. Appl. Math. 32:588–597 (1977)]. Explicit results using the method are reported to O(ε) and confirmed with asymptotic expansions of the exact solution; the O(1) results agree with those reported in the literature. In the case where the coefficient functions are analytic, an exact solution is presented. It is not difficult to perform the higher-order calculations using the proposed variational approach, in contrast to the current methods in use.     

Convergence of cascade algorithms introduced by I. Daubechies

Orthonormal wavelets are often constructed by an iteration procedure commonly referred to as thecascade algorithm. We obtain an asymptotic expansion and deduce necessary and sufficient conditions for convergence in Sobolev norms.     

A Small Noise Asymptotic Expansion for Young SDE Driven by Fractional Brownian Motion: A Sharp Error Estimate With Malliavin Calculus

This article shows an analytically tractable small noise asymptotic expansion with a sharp error estimate for the expectation of the solution to Young’s pathwise stochastic differential equations (SDEs) driven by fractional Brownian motions with the Hurst index H > 1/2. In particular, our asymptotic expansion can be regarded as small noise and small time asymptotics by the error estimate with Malliavin culculus. As an application, we give an expansion formula in one-dimensional general Young SDE driven by fractional Brownian motion. We show the validity of the expansion through numerical experiments.     

Approximations to the distribution of the sample correlation matrix

In this article, multivariate density expansions for the sample correlation matrix R are derived. The density of R is expressed through multivariate normal and through Wishart distributions. Also, an asymptotic expansion of the characteristic function of R is derived and the main terms of the first three cumulants of R are obtained in matrix form. These results make it possible to obtain asymptotic density expansions of multivariate functions of R in a direct way.     

Formal asymptotic solution of a singularly perturbed nonlinear optimal control problem

A system of equations that arises in a singularly perturbed optimal control problem is studied. We give conditions under which a formal asymptotic solution exists. This formal asymptotic solution consists of an outer expansion and left and right boundary-layer expansions. A feature of our procedure is that we do nota priori eliminate the control function from the problem. In particular, we construct a formal asymptotic expansion for the control directly. We apply our procedure to a Mayer-type problem. The paper concludes with a worked example.     

Asymptotics for M/G/1 low-priority waiting-time tail probabilities

We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptive-resume disciplines. We show that the low-priority steady-state waiting-time can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waiting-time distribution. We exploit this structures to determine the asymptotic behavior of the tail probabilities. Unlike the FIFO case, there is routinely a region of the parameters such that the tail probabilities have non-exponential asymptotics. This phenomenon even occurs when both service-time distributions are exponential. When non-exponential asymptotics holds, the asymptotic form tends to be determined by the non-exponential asymptotics for the high-priority busy-period distribution. We obtain asymptotic expansions for the low-priority waiting-time distribution by obtaining an asymptotic expansion for the busy-period transform from Kendall's functional equation. We identify the boundary between the exponential and non-exponential asymptotic regions. For the special cases of an exponential high-priority service-time distribution and of common general service-time distributions, we obtain convenient explicit forms for the low-priority waiting-time transform. We also establish asymptotic results for cases with long-tail service-time distributions. As with FIFO, the exponential asymptotics tend to provide excellent approximations, while the non-exponential asymptotics do not, but the asymptotic relations indicate the general form. In all cases, exact results can be obtained by numerically inverting the waiting-time transform. This revised version was published online in June 2006 with corrections to the Cover Date.