Statistical inference for quantiles in the frequency domain

For second-order stationary processes, the spectral distribution function is uniquely determined by the autocovariance function of the process. We define the quantiles of the spectral distribution function in frequency domain. The estimation of quantiles for second-order stationary processes is considered by minimizing the so-called check function. The quantile estimator is shown to be asymptotically normal. We also consider a hypothesis testing for quantiles in frequency domain and propose a test statistic associated with our quantile estimator, which asymptotically converges to standard normal under the null hypothesis. The finite sample performance of the quantile estimator is shown in our numerical studies.     

CONVERGENCE OF THE CLIPPED SAMPLE AUTOCORRELATION AND AUTOCOVARIANCE

1.IntroductionClassicalmethodstoestimatetheautocorrelationandautocovarianceofastationarylineartimeseriesworkwellforgoodobservations,buttheyaresensitivetotheexistenceofoutliers.[1]gayeamethodofestimatinganARprocessbycountingthenumberofcertainrunsintheassociatedclippedbinaryseries.Althoughinsodoingsomeefficiencyislost,theproceduregivesgoodestimatesagainstoutliers.LetX.,n=0,11,12,'1beareallinearGaussiansequencewithzeromeanandautocorrelationspj~EXoXj/EXI,j=0,1,'landXu,n~0,11,',theclipped…     

Statistical Properties of Covariance Tapers

Compactly supported autocovariance functions reduce computations needed for estimation and prediction under Gaussian process models, which are commonly used to model spatial and spatial-temporal data. A critical issue in using such models is the loss in statistical efficiency caused when the true autocovariance function is not compactly supported. Theoretical results indicate the value of specifying the local behavior of the process correctly. One way to obtain a compactly supported autocovariance function that has similar local behavior to an autocovariance function K of interest is to multiply K by some smooth compactly supported autocovariance function, which is called covariance tapering. This work extends previous theoretical results showing that covariance tapering has some asymptotic optimality properties as the number of observations in a fixed and bounded domain increases. However, numerical experiments show that for purposes of parameter estimation, covariance tapering often does not work as well as the simple alternative of breaking the observations into blocks and ignoring dependence across blocks. When covariance tapering is used for spatial prediction, predictions near the boundary of the observation domain are affected most. This article proposes an approach to modifying the taper to ameliorate this edge effect. In addition, a justification for a specific approach to carrying out conditional simulations based on tapered covariances is given. Supplementary materials for this article are available online.     

Safety Stock Determination with Serially Correlated Demand in a Periodic-Review Inventory System

We consider a periodic-review inventory replenishment model with an order-up-to-R operating doctrine for the case of deterministic lead times and a covariance-stationary stochastic demand process. A method is derived for setting the inventory safety stock to achieve an exact desired stockout probability when the autocovariance function for Gaussian demand is known. Because the method does not require that parametric time-series models be fit to the data, it is easily implemented in practice. Moreover, the method is shown to be asymptotically valid when the autocovariance function of demand is estimated from historical data. The effects on the stockout rate of various levels of autocorrelated demand are demonstrated for situations in which autocorrelation in demand goes undetected or is ignored by the inventory manager. Similarly, the changes to the required level of safety stock are demonstrated for varying levels of autocorrelation.     

Diagnostic checking of multivariate nonlinear time series models with martingale difference errors

In this article, we derive the asymptotic distribution of residual autocovariance and autocorrelation matrices for a general class of multivariate nonlinear time series models by assuming only that the error term is a martingale difference sequence. Two types of applications are developed: global test statistics of the portmanteau type and one-lag test statistics, which describe the residual correlation at individual lags. To illustrate the proposed methodology, simulation results are reported for diagnosing multivariate threshold time series models. The following test statistics are compared: the classical test statistics presuming independent errors and the proposed methodology which supposes only martingale difference errors.     

独立非同分布二元高斯三角阵最大值的渐近性及相关统计推断

考虑二元独立非同分布高斯随机向量三角阵列最大值分布的渐近性及相关统计推断.此高斯三角阵的第n列的第i个向量服从二元高斯分布,其相关系数为i/n的函数并单调连续.首先建立了此高斯三角阵最大值分布的一阶和二阶渐近展开式.其次,分析相关系数参数估计及估计量的渐近性质.最后,通过随机模拟说明了相关系数之参数估计的有效性,并将该二元非同分布三角阵列模型应用于实际数据,得到了满意的结果.     

Gaussian Scenario for the Heat Equation with Quadratic Potential and Weakly Dependent Data with Applications

For a suitable scaling of the solution to the one-dimensional heat equation with spatial-dependent coefficients and weakly dependent random initial conditions, the convergence to the Gaussian limiting distribution is proved. The scaling proposed and methodology followed allow us to obtain Gaussian scenarios for related equations such as the one-dimensional Burgers equation as well as for the multidimensional formulation of both the heat and Burgers equations. Furthermore, the investigation of non-Gaussian scenarios is opened with a different proposed scaling, proving the convergence of the second-order moments.      

A HYBRID EXPLICIT-IMPLICIT SCHEME FOR THE TIME-DEPENDENT WIGNER EQUATION

This paper designs a hybrid scheme based on finite difference methods and a spectral method for the time-dependent Wigner equation,and gives the error analysis for the full discret ization of its initial value problem.An explicit-implicit time-splitting scheme is used for time integration and the second-order upwind finite difference scheme is used to dis-cretize the advection term.The consistence error and the stability of the full discretization are analyzed.A Fourier spectral method is used to approximate the pseudo-differential operator term and the corresponding error is studied in detail.The final convergence result shows clearly how the regularity of the solution affects the convergence order of the pro-posed scheme.N umerical results are presented for confirming the sharpness of the analysis.The scattering effects of a Gaussian wave packet tunneling through a Gaussian potential barrier are investigated.The evolution of the density function shows that a larger portion of the wave is reflected when the height and the width of the barrier increase.Mathematics subject classification:65M06,65M70.     

一类Gauss序列极值的极限律

本文讨论一类非平稳Gauss序列的极值．利用点过程收敛定理得到多水平超过的点过程的收敛性,同时得到在不相交区间上最大值的联合渐近分布,第k个最大值的渐近分布以及前r个极值的联合渐近状态．     

Stein shrinkage and second-order efficiency for semiparametric estimation of the shift

The problem of estimating the shift (or, equivalently, the center of symmetry) of an unknown symmetric and periodic function f observed in Gaussian white noise is considered. Using the blockwise Stein method, a penalized profile likelihood with a data-driven penalization is introduced so that the estimator of the center of symmetry is defined as the maximizer of the penalized profile likelihood. This estimator has the advantage of being independent of the functional class to which the signal f is assumed to belong and, furthermore, is shown to be semiparametrically adaptive and efficient. Moreover, the second-order term of the risk expansion of the proposed estimator is proved to behave at least as well as the second-order term of the risk of the best possible estimator using monotone smoothing filter. Under mild assumptions, this estimator is shown to be second-order minimax sharp adaptive over the whole scale of Sobolev balls with smoothness β > 1. Thus, these results extend those of [10], where second-order asymptotic minimaxity is proved for an estimator depending on the functional class containing f and β ≥ 2 is required.      

L p -Loss and Limit Distribution for Predicting Integrals of Some Non-Gaussian Second Order Processes

Mean squared error for estimating integrals of second order processes like Fractional Brownian Motion have been considered since a while. If the process under study is Gaussian, the distribution of the estimator and its L ^p -loss can be easily deduced. The aim of this paper is to exhibit a class of non-Gaussian second-order processes, for which the distribution of the estimator and its L ^p -loss differ dramatically from the Gaussian case. This revised version was published online in August 2006 with corrections to the Cover Date.     

Tracy-Widom distribution,Airy2 process and its sample path properties

Tracy-Widom distribution was first discovered in the study of largest eigenvalues of high dimensional Gaussian unitary ensembles(GUE),and since then it has appe...     

On the efficiency of Gaussian adaptation

Gaussian Adaptation (GA) is a stochastic process that adapts a Gaussian distribution to a region or set of feasible points in parameter space. As a result of the adaptation, GA becomes a maximum dispersion process extending the sampling over the largest possible volume in parameter space while keeping the probability of finding feasible points at a suitable level. For such a process, a general measure of efficiency is defined and an efficiency theorem is proved.     

Remarks on limit theorems for nonlinear functionals of Gaussian sequences

Summary Limit theorems for sums of nonlinear functionals of Gaussian sequences typically obtain as limit distribution that of a single term in an expansion given by Dobrushin [1] for a process subordinate to a Gaussian process. Here we show how one can obtain limit theorems of this type where the limit distribution is that of a full expansion of Dobrushin's type.This research is supported in part by Office of Naval Research contract N00014-81-K-003 and National Science Foundation Grant No. DMS 83-12106     

On the maximal domain of attraction of Tracy–Widom distribution for Gaussian unitary ensembles

The asymptotic distribution of the largest eigenvalue of various classes of random matrices has been shown to have the Tracy–Widom distribution. In this article, we prove that the standardised maximum of an independent and identically distributed sequence of random variables having the Tracy–Widom distribution arising from the Gaussian unitary ensemble belongs to the Gumbel domain of attraction.     

Some test statistics based on the martingale term of the empirical distribution function

Summary It is proved that the martingale term of the empirical distribution function converges weakly to a Gaussian process inD[0, 1]. Some statistics for goodness-of-fit tests based on the martingale term of the empirical distribution function are proposed. Asymptotic distributions of these statistics under the null hypothesis are given. The approximate Bahadur efficiencies of the statistics to the Kolmogorov-Smirnov statistic and to the Cramér-von Mises statistic are also calculated. The Institute of Statistical Mathematics     

Optimal estimation of a signal perturbed by a sub-fractional Brownian motion

We consider the problem of optimal estimation of the vector parameter θ of the drift term in a sub-fractional Brownian motion. We obtain the maximum likelihood estimator as well as Bayesian estimator when the prior distribution is Gaussian.     

A second-order Monte Carlo method for the solution of the Ito stochastic differential equation

A difference approximation that is second-order accurate in the time step his derived for the general Ito stochastic differential equation. The difference equation has the form of a second-order random walk in which the random terms are non-linear combinations of Gaussian random variables. For a wide class of problems, the transition pdf is joint-normal to second order in h; the technique then reduces to a Gaussian random walk, but its application is not limited to problems having a Gaussian solution. A large number of independent sample paths are generated in a Monte Carlo solution algorithm; any statistical function of the solution (e.g., moments or pdf's) can be estimated by ensemble averaging over these paths     

Second order tail asymptotics for the sum of dependent, tail-independent regularly varying risks

In this paper we consider dependent random variables with common regularly varying marginal distribution. Under the assumption that these random variables are tail-independent, it is well known that the tail of the sum behaves like in the independence case. Under some conditions on the marginal distributions and the dependence structure (including Gaussian copula’s and certain Archimedean copulas) we provide the second-order asymptotic behavior of the tail of the sum.     

The empirical distribution function and partial sum process of residuals from a stationary arch with drift process

The weak convergence of the empirical process and partial sum process of the residuals from a stationary ARCH-M model is studied. It is obtained for and consistent estimate of the ARCH-M parameters. We find that the limiting Gaussian processes are no longer distribution free and hence residuals cannot be treated as i.i.d. In fact the limiting Gaussian process for the empirical process is a standard Brownian bridge plus an additional term, while the one for partial sum process is a standard Brownian motion plus an additional term. In the special case of a standard ARCH process, that is an ARCH process with no drift, the additional term disappears. We also study a sub-sampling technique which yields the limiting Gaussian processes for the empirical process and partial sum process as a standard Brownian bridge and a standard Brownian motion respectively.