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.     

An Asymptotic Expansion for the Distribution of Hotelling'sT-Statistic under Nonnormality

In this paper we obtain an asymptotic expansion for the distribution of Hotelling'sT²-statisticT²under nonnormality when the sample size is large. In the derivation we find an explicit Edgeworth expansion of the multivariatet-statistic. Our method is to use the Edgeworth expansion and to expand the characteristic function ofT².     

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.     

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.     

On a test of equality of two-parameter exponential distributions

In this paper, an asymptotic expansion of the distribution of the statistic for testing the equality of p two-parameter exponential distributions is obtained upto the order n^?4 with the second term of the order n^?3 where n is the size of the sample drawn from the i th exponential population. The asymptotic expansion can therefore be used to obtain accurate approximations to the critical values of the test statistic even for comparatively small values of n. Also we have shown that F-tables can be used to test the hypothesis when the sample size is moderately large.     

Resurgent aspects of applied exponential asymptotics

In many physical problems, it is important to capture exponentially small effects that lie beyond-all-orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans-series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans-series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans-series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial-over-power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial-over-power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences.     

The asymptotics of the solutions of the signorini problem without friction or with small friction

We find the first few terms of the asymptotic expansion of a regular solution of the two-dimensional Signorini problem with a small coefficient of friction. As the fundamental approximation we take the solution of the limiting problem without friction. This solution is assumed to be known, and it is assumed that the region of contact consists of a finite number of arcs, on each of which one boundary condition or another is realized. We study the asymptotics of the solution of the Signorini problem without friction under small load variation. Bibliography: 12 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 82–110.     

Asymptotics of Solutions of the Einstein Equations with Positive Cosmological Constant

A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.Communicated by Sergiu Klainermansubmitted 08/12/03, accepted 30/04/04     

High-dimensional asymptotic expansion of LR statistic for testing intraclass correlation structure and its error bound

This paper deals with the null distribution of a likelihood ratio (LR) statistic for testing the intraclass correlation structure. We derive an asymptotic expansion of the null distribution of the LR statistic when the number of variable p and the sample size N approach infinity together, while the ratio p/N is converging on a finite nonzero limit c(0,1). Numerical simulations reveal that our approximation is more accurate than the classical χ²-type and F-type approximations as p increases in value. Furthermore, we derive a computable error bound for its asymptotic expansion.     

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.     

Tail Asymptotics for HOL Priority Queues Handling a Large Number of Independent Stationary Sources

In this paper we study the asymptotics of the tail of the buffer occupancy distribution in buffers accessed by a large number of stationary independent sources and which are served according to a strict HOL priority rule. As in the case of single buffers, the results are valid for a very general class of sources which include long-range dependent sources with bounded instantaneous rates. We first consider the case of two buffers with one of them having strict priority over the other and we obtain asymptotic upper bound for the buffer tail probability for lower priority buffer. We discuss the conditions to have asymptotic equivalents. The asymptotics are studied in terms of a scaling parameter which reflects the server speed, buffer level and the number of sources in such a way that the ratios remain constant. The results are then generalized to the case of M buffers which leads to the source pooling idea. We conclude with numerical validation of our formulae against simulations which show that the asymptotic bounds are tight. We also show that the commonly suggested reduced service rate approximation can give extremely low estimates.     

Analytic combinatorics of connected graphs

We enumerate the connected graphs that contain a number of edges growing linearly with respect to the number of vertices. So far, only the first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, that is, generating function manipulations, we derive a formula for the coefficients of the complete asymptotic expansion. The same result is derived for connected multigraphs.     

On the internal path length of d‐dimensional quad trees

It is proved that the internal path length of a d‐dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the m‐ary search trees. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 5: 25–41, 1999     

Self–similarity and instability in classical mean–field models for domain coarsening

In the classical theory by Lifshitz, Slyozov and Wagner (LSW) coarsening of a dilute system of particles is modelled by a nonlocal transport equation for the particle size distribution. LSW predict that the asymptotic behavior for large times is self–similar and that a particular self–similar profile is approached. In this talk we discuss rigorous results on the long–time behavior of solutions for several variants of this model. For systems in which particle size is uniformly bounded these results establish a sensitive dependence on the data and thus in general do not confirm the predictions by LSW. More precisely we prove that convergence to the classically predicted similarity solution is impossible if the initial distribution is comparable to any finite power of distance to the end of the support. In addition we give a necessary criterion for convergence to other self–similar solutions, which implies non–self–similar asymptotics for a dense set of data.     

Nonlinear wavelet density estimation with censored dependent data

In this paper, we provide an asymptotic expansion for the mean integrated squared error (MISE) of nonlinear wavelet estimator of survival density for a censorship model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary and α‐mixing sequence. This asymptotic MISE expansion, when the density is only piecewise smooth, is same. However, for the kernel estimators, the MISE expansion fails if the additional smoothness assumption is absent. Also, we establish the asymptotic normality of the nonlinear wavelet estimator. Copyright © 2011 John Wiley & Sons, Ltd.     

Limiting behaviour of the mean residual life

In survival or reliability studies, the mean residual life or life expectancy is an important characteristic of the model. Here, we study the limiting behaviour of the mean residual life, and derive an asymptotic expansion which can be used to obtain a good approximation for large values of the time variable. The asymptotic expansion is valid for a quite general class of failure rate distributions—perhaps the largest class that can be expected given that the terms depend only on the failure rate and its derivatives.     

The Asymmetric Leader Election Algorithm: Another Approach

The asymmetric leader election algorithm has obtained quite a bit of attention lately. In this paper we want to analyze the following asymptotic properties of the number of rounds: Limiting distribution function, all moments in a simple automatic way, asymptotics for p → 0, p → 1 (where p denotes the “killing” probability). This also leads to a few interesting new identities. We use two paradigms: First, in some urn model, we have asymptotic independence of urns behaviour as far as random variables related to urns with a fixed number of balls are concerned. Next, we use a technique easily leading to the asymptotics of the moments of extremevalue related distribution functions.      

Weak convergence of random <Emphasis Type="Bold">p</Emphasis>-mappings and the exploration process of inhomogeneous continuum random trees

We study the asymptotics of the p-mapping model of random mappings on [n] as n gets large, under a large class of asymptotic regimes for the underlying distribution p. We encode these random mappings in random walks which are shown to converge to a functional of the exploration process of inhomogeneous random trees, this exploration process being derived (Aldous-Miermont-Pitman 2004) from a bridge with exchangeable increments. Our setting generalizes previous results by allowing a finite number of “attracting points” to emerge.Research supported by NSF Grant DMS-0203062.Research supported by NSF Grant DMS-0071468.     

Asymptotics for multiple Meixner polynomials

We study the asymptotic behavior of multiple Meixner polynomials of first and second kind, respectively [6]. We use an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution. Moreover, analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics.     

Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process

Discrete random probability measures are a key ingredient of Bayesian nonparametric inference. A sample generates ties with positive probability and a fundamental object of both theoretical and applied interest is the corresponding number of distinct values. The growth rate can be determined from the rate of decay of the small frequencies implying that, when the decreasingly ordered frequencies admit a tractable form, the asymptotics of the number of distinct values can be conveniently assessed. We focus on the geometric stick-breaking process and we investigate the effect of the distribution for the success probability on the asymptotic behavior of the number of distinct values. A whole range of logarithmic behaviors are obtained by appropriately tuning the prior. A two-term expansion is also derived and illustrated in a comparison with a larger family of discrete random probability measures having an additional parameter given by the scale of the negative binomial distribution.