Spectral density estimation with amplitude modulation and outlier detection

This paper studies spectral density estimation based on amplitude modulation including missing data as a specific case. A generalized periodogram is introduced and smoothed to give a consistent estimator of the spectral density by running local linear regression smoother. We explore the asymptotic properties of the proposed estimator and its application to time series data with periodic missing. A simple data-driven local bandwidth selection rule is proposed and an algorithm for computing the spectral density estimate is presented. The effectiveness of the proposed method is demonstrated using simulations. The application to outlier detection based on leave-one-out diagnostic is also considered. An illustrative example shows that the proposed diagnostic procedure succeeds in revealing outliers in time series without masking and smearing effects. Supported by Chinese NSF Grants 10001004 and 39930160, and Fellowship of City University of Hong Kong.     

BANDWIDTH SELECTION IN NONPARAMETRIC SPECTRAL DENSITY ESTIMATION OF THE STATIONARY GAUSSIAN PROCESS

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On the Estimation of the Parameters of Multivariate Stable Distributions

In the paper, the asymptotic normality for a new estimator for the spectral measure of a multivariate stable distribution is proved. Also an estimator for the density of a multivariate stable distribution is proposed, its properties are investigated. The dependence of a stable density on exponent and the spectral measure is investigated.     

Asymptotic normality of spectral estimates

The asymptotic normality of some spectral estimates, including a functional central limit theorem for an estimate of the spectral distribution function, is proved for fourth-order stationary processes. In contrast to known results it is not assumed that all moments exist or that the process is linear. The data are allowed to be tapered. Using some recent results on the central limit theorem for stationary processes, corollaries are obtained for strong and φ-mixing sequences and linear transformations of martingale differences.     

Asymptotic estimates of multi-dimensional stable densities and their applications

The relation between the upper and lower asymptotic estimates of the density and the fractal dimensions on the sphere of the spectral measure for a multivariate stable distribution is discussed. In particular, the problem and the conjecture on the asymptotic estimates of multivariate stable densities in the work of Pruitt and Taylor in 1969 are solved. The proper asymptotic orders of the stable densities in the case where the spectral measure is absolutely continuous on the sphere, or discrete with the support being a finite set, or a mixture of such cases are obtained. Those results are applied to the moment of the last exit time from a ball and the Spitzer type limit theorem involving capacity for a multi-dimensional transient stable process.

     

Screening Effect in Isotropic Gaussian Processes

The screening effect is the phenomenon that optimal linear prediction of a spatial process on an unobserved location mostly depends on nearby observations.That is,the optimal predictor based on just nearby observations yields a good approximation of which based on the whole large dataset.However,the approximation does not always perform well since the screening effect may not hold in all situations.To determine when the screening effect holds is an important issue in spatial statistics.This paper provides some sufficient conditions for ensuring an asymptotic screening effect in Rd based on the spectral density of the underlying isotropic Gaussian process and the geometries of nearby observations.These results apply to isotropic processes with an arbitrary degree of differentiability.Assuming we are predicting at origin,the conditions are(1)the spectral density is nearly a constant in balls of finite radius far from the origin,(2)the positions of nearby observations do not fall on a curve with non-zero intercept whose degree is less or equal to the order of mean square differentiability of the process.These conditions are easy to verify in practice.Convergence rates of the asymptotic screening effect are also obtained.These rates depend on the rate of decrease of the spectral density.Simulation studies on the screening effect for finite samples are also provided.     

Critical Asymptotic Behavior of Correlations in a Simple Liquid

We consider a representation of the total and the direct correlation functions in a liquid in terms of the corresponding spectral densities. Analysis of the spectral density properties reveals a mechanism responsible for changing the analytic properties of the Fourier transform of the correlation function and its asymptotic behavior in the critical domain. We show that the well-known restrictions imposed on possible values of the critical asymptotic exponent follow from this mechanism.     

Testing a hypothesis about the spectrum of a linearly regular sequence

We study the asymptotic behavior of the maximum deviation of the estimate for the spectral density of the form (2) from the true value of the spectral density of a linearly regular sequence and propose some methods of constructing confidence intervals and testing hypotheses concerning this spectral density. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez. pp. 10–15, Perm, 1993.     

BOOTSTRAP MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETER IN SPECTRAL DENSITY OF STATIONARY PROCESSES

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可修复人机系统的指数稳定性

研究了两相同部件温储备可修的人机系统,运用C_0半群的相关理论,对系统主算子的谱界进行估值.估算系统的算子产生的半群的增长界,然后运用了共尾的概念及相关的理论,得到了系统算子A+B的谱界与系统算子产生的半群的增长界相同.进而运用相关代数知识证得,0为系统算子的简单本征值,并分析了系统算子的谱分布,得到系统的指数稳定性.并研究了系统算子预解式的特性.对任意给定的δ0,γ=a+bi,-μ+δa_1≤a≤a_2,得到lim|b|→∞‖R(γ;A+B)‖=0.进而得到在~sRγ≥a_1的右半平面内相应于系统算子A+B的谱点由有限个本征值组成.     

Asymptotic behavior of spectral of Neumann‐Poincaré operator in Helmholtz system

In this paper, we are concerned with the asymptotic behavior of the Neumann‐Poincaré operator in Helmholtz system. By analyzing the asymptotic behavior of spherical Bessel function near the origin and/or approach higher order, we prove the asymptotic behavior of spectral of Neumann‐Poincaré operator when frequency is small enough and/or the order is large enough. The results show that spectral of Neumann‐Poincaré operator is continuous at the origin and converges to 0 from the complex plane in general.     

一类具有备用部件的可修人机系统解的渐近稳定性

运用泛函分析的方法 ,通过分析系统主算子的谱特征 ,给出一类具有备用部件的可修人机系统解的渐近稳定性证明 .     

平稳随机过程的小波分析方法

利用小波变换对平稳随机过程进行了谱分析,在小波变换的基础上给出了平稳随机过程的时—频功率谱及联合平稳随机过程的时—频互功率谱的概念,并详尽地研究了它们所具有的性质及与传统功率谱的关系。     

Asymptotic spectral analysis of growing regular graphs

We propose the quantum probabilistic techniques to obtain the asymptotic spectral distribution of the adjacency matrix of a growing regular graph. We prove the quantum central limit theorem for the adjacency matrix of a growing regular graph in the vacuum and deformed vacuum states. The condition for the growth is described in terms of simple statistics arising from the stratification of the graph. The asymptotic spectral distribution of the adjacency matrix is obtained from the classical reduction.

     

Electromagnetic scattering by a smooth convex impedance cone

The problem of the diffraction of an electromagnetic planewave by a convex cone of arbitrary smooth cross-section withimpedance (Leontovich) boundary conditions is studied. The vectorproblem is reduced to that for the Debye potentials. By meansof Kontorovich–Lebedev integrals, two spectral functionsare introduced and the corresponding boundary value problemis formulated. The spectral functions for the potentials arefound to satisfy the Helmholtz equations on the unit sphereand to be coupled through non-traditional boundary conditionsof the impedance type with shifts on the spectral variable.The use of the Green theorem permits us to establish an integralformulation of the boundary value problem for the spectral functions.The formal asymptotic solution of the problem is then givenfor the case of a narrow cone. For this, two different methodsare given: a method of perturbation applied to the spectralintegral equations and an adaptation of the method of matchingthe asymptotic series in spectral domain. Both methods leadto the same closed-form result for the leading term of the scatteringdiagram asymptotics.     

Asymptotics near the boundary of spectral functions of elliptic self-adjoint boundary problems

We obtain the asymptotic behavior of the spectral function of an elliptic self-adjoint boundary problem in a closed domain and give a uniform estimate for the remainder. The estimate for the remainder is the same as the interior estimates of Agmon-Kannai and Hormander.     

A functional limit theorem for tapered empirical spectral functions

Using convolution properties of frequency-kernels and their upper bounds we obtain some new upper bounds for the cumulants of time series statistics. From these results we derive the asymptotic normality of some spectral estimates and the tightness of tapered empirical spectral functions in the space of Lipschitz-continuous functions. It follows that tapering increases the asymptotic variance of the estimates by a constant factor. All results are proved under integrability conditions on the spectra. A functional limit theorem for the empirical spectral function is also given without assuming all moments of the underlying process to exist.     

Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients

We consider a non-self-adjoint Schrödinger operator describing the motion of a particle in a one-dimensional space with an analytic potential iV (x) that is periodic with a real period T and is purely imaginary on the real axis. We study the spectrum of this operator in the semiclassical limit and show that the points of its spectrum asymptotically belong to the so-called spectral graph. We construct the spectral graph and evaluate the asymptotic form of the spectrum. A Riemann surface of the particle energy-conservation equation can be constructed in the phase space. We show that both the spectral graph and the asymptotic form of the spectrum can be evaluated in terms of integrals of the pdx form (where x ∈31 ?/T? and p ∈, ? are the particle coordinate and momentum) taken along basis cycles on this Riemann surface. We use the technique of Stokes lines to construct the asymptotic form of the spectrum.     

方差分量的广义谱分解估计

对于随机效应部分为一般平衡多向分类的线性混合模型,将王松桂(2002)提出的一种称之为谱分解估计的参数估计新方法推广到随机效应设计阵为任意矩阵的含两个方差分量的线性混合模型,给出了方差分量的广义谱分解估计方法,并证明了所得估计的一些统计性质。另外,还就广义谱分解估计类中某些特殊估计和对应的方差分析估计进行了比较,得到了它们相等的充分必要条件。     

Variational principle and asymptotic behavior of the density of states for random pseudodifferential operators

A variational principle (similar to Glazman's lemma) is proved for the density of the states of random self-adjoint hypoelliptic operators (of positive or negative order). Some results from the index theory of random elliptic operators are used in the proof. As an application, the asymptotic behavior of the density of the states of such operators and an estimate of the remainder are obtained by the method of the approximate spectral projection.The authors are grateful to V. I. Bezyaev for numerous useful discussions.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 98–117, 1986.