On the theory of continuous time series

This paper considers spectral estimation for a zero-mean strictly stationary r-vector valued continuous time series. The case of interest is when some of observations are missing due to some random failure. Spectral estimation procedures are developed in disjoint segments of observations. Expanded finite Fourier transform, modified poriodogram and spectral density statistics are constructed. The theoretical properties of these estimators are developed. Asymptotic distributions are discussed     

An Approximation Method for Improving Dynamic Network Model Fitting

There has been a great deal of interest recently in the modeling and simulation of dynamic networks, that is, networks that change over time. One promising model is the separable temporal exponential-family random graph model (ERGM) of Krivitsky and Handcock, which treats the formation and dissolution of ties in parallel at each time step as independent ERGMs. However, the computational cost of fitting these models can be substantial, particularly for large, sparse networks. Fitting cross-sectional models for observations of a network at a single point in time, while still a nonnegligible computational burden, is much easier. This article examines model fitting when the available data consist of independent measures of cross-sectional network structure and the duration of relationships under the assumption of stationarity. We introduce a simple approximation to the dynamic parameters for sparse networks with relationships of moderate or long duration and show that the approximation method works best in precisely those cases where parameter estimation is most likely to fail—networks with very little change at each time step. We consider a variety of cases: Bernoulli formation and dissolution of ties, independent-tie formation and Bernoulli dissolution, independent-tie formation and dissolution, and dependent-tie formation models.     

Image reconstruction in multi-channel model under Gaussian noise

The image reconstruction from noisy data is studied. A nonparametric boundary function is estimated from observations in a growing number, N, of independent channels in the Gaussian white noise. In each channel, the image and the background intensities are unknown. They define a set of unidentifiable nuisance parameters that slow down the typical minimax rate of convergence. The asymptotically minimax rate is found as N → ∞, and an asymptotically optimal estimator of the boundary function is suggested.      

Minimax estimators of spectral intensities of acoustic sources

We investigate some issues of minimax estimation of spectral intensity of acoustic sources in cases when the noise and the spectral amplitudes satisfy quadratic constraints. Recursive minimax filters are obtained.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 116–119, 1986.     

The root–unroot algorithm for density estimation as implemented via wavelet block thresholding

We propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem. This regression problem is created by binning the original observations into many small size bins, and by then applying a suitable form of root transformation to the binned data counts. In principle many common nonparametric regression estimators could then be applied to the transformed data. We propose use of a wavelet block thresholding estimator in this paper. Finally, the estimated regression function is un-rooted by squaring and normalizing. The density estimation procedure achieves simultaneously three objectives: computational efficiency, adaptivity, and spatial adaptivity. A numerical example and a practical data example are discussed to illustrate and explain the use of this procedure. Theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. There are three key steps in the technical argument: Poissonization, quantile coupling, and oracle risk bound for block thresholding in the non-Gaussian setting. Some of the technical results may be of independent interest.     

A nonlinear time series model and estimation of missing observations

This paper formulates a nonlinear time series model which encompasses several standard nonlinear models for time series as special cases. It also offers two methods for estimating missing observations, one using prediction and fixed point smoothing algorithms and the other using optimal estimating equation theory. Recursive estimation of missing observations in an autoregressive conditionally heteroscedastic (ARCH) model and the estimation of missing observations in a linear time series model are shown to be special cases. Construction of optimal estimates of missing observations using estimating equation theory is discussed and applied to some nonlinear models.Authors were supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.     

The stochastic generalised assignment problem with Bernoulli demands

This paper deals with the Stochastic Generalised Assignment problem. It presents several models for the special case when demands are independent and Bernoulli distributed. Each model designs an assignment structure before the demands are known. Two policies are considered to handle infeasibilities in particular instances of the demands vector. Model performances are compared under both policies.     

Nonparametric inference for the spectral measure of a bivariate pure-jump semimartingale

We develop a nonparametric estimator for the spectral density of a bivariate pure-jump Itô semimartingale from high-frequency observations of the process on a fixed time interval with asymptotically shrinking mesh of the observation grid. The process of interest is locally stable, i.e., its Lévy measure around zero is like that of a time-changed stable process. The spectral density function captures the dependence between the small jumps of the process and is time invariant. The estimation is based on the fact that the characteristic exponent of the high-frequency increments, up to a time-varying scale, is approximately a convolution of the spectral density and a known function depending on the jump activity. We solve the deconvolution problem in Fourier transform using the empirical characteristic function of locally studentized high-frequency increments and a jump activity estimator.     

Nonparametric regression with warped wavelets and strong mixing processes

We consider the situation of a univariate nonparametric regression where either the Gaussian error or the predictor follows a stationary strong mixing stochastic process and the other term follows an independent and identically distributed sequence. Also, we estimate the regression function by expanding it in a wavelet basis and applying a hard threshold to the coefficients. Since the observations of the predictor are unequally distant from each other, we work with wavelets warped by the density of the predictor variable. This choice enables us to retain some theoretical and computational properties of wavelets. We propose a unique estimator and show that some of its properties are the same for both model specifications. Specifically, in both cases the coefficients are unbiased and their variances decay at the same rate. Also the risk of the estimator, measured by the mean integrated square error is almost minimax and its maxiset remains unaltered. Simulations and an application illustrate the similarities and differences of the proposed estimator in both situations.

     

Minimax estimation of a variance

The nonparametric problem of estimating a variance based on a sample of sizen from a univariate distribution which has a known bounded range but is otherwise arbitrary is treated. For squared error loss, a certain linear function of the sample variance is seen to be minimax for eachn from 2 through 13, exceptn=4. For squared error loss weighted by the reciprocal of the variance, a constant multiple of the sample variance is minimax for eachn from 2 through 11. The least favorable distribution for these cases gives probability one to the Bernoulli distributions.     

Periodogram analysis with missing observations

Estimation of the spectral measure, covariance and spectral density functions of a strictly stationaryr- vector valued time series is considered, under the assumption that some of the observations are missed. The modified periodograms are calculated using data window. The asymptotic normality is studied.     

On Bernoulli decompositions for random variables, concentration bounds, and spectral localization

As was noted already by A. N. Kolmogorov, any random variable has a Bernoulli component. This observation provides a tool for the extension of results which are known for Bernoulli random variables to arbitrary distributions. Two applications are provided here: (i) an anti-concentration bound for a class of functions of independent random variables, where probabilistic bounds are extracted from combinatorial results, and (ii) a proof, based on the Bernoulli case, of spectral localization for random Schrödinger operators with arbitrary probability distributions for the single site coupling constants. For a general random variable, the Bernoulli component may be defined so that its conditional variance is uniformly positive. The natural maximization problem is an optimal transport question which is also addressed here.     

时间序列中方差的结构变点的小波识别(英文)

本文给出了时间序列中方差的小波系数的两种估计:连续估计和离散估计.这两种估计可以用来检测时间序列中方差的结构变点.利用这两种估计我们给出了方差变点的位置和跳跃幅度的估计,并且显示出这些估计可达到最佳收敛速度.同时,我们还给出了这些估计的收敛速度以及检验统计量的渐进分布!     

下三角双线性时间序列模型的近似三阶矩结构和双谱密度

下三角双线性时间序列模型，特别是它的一些简单的特殊情况，被许多人研究过。但对于一般形式，目前，只知道其二阶结构(自协方差和谱)与线性ARMA模型相似.而反映该模型特征的三阶结构(三阶矩和双谱),由于既繁琐又复杂而很难获得(文献中尚未见报道）。该文给出一种计算三阶矩和双谱的 近似方法。特别地，对于可分离的下三角双线性时间序列模型，得到了比较简洁实用的计算公式。     

Minimax regret solution to multiobjective linear programming problems with interval objective functions coefficients

The current paper focuses on a multiobjective linear programming problem with interval objective functions coefficients. Taking into account the minimax regret criterion, an attempt is being made to propose a new solution i.e. minimax regret solution. With respect to its properties, a minimax regret solution is necessarily ideal when a necessarily ideal solution exists; otherwise it is still considered a possibly weak efficient solution. In order to obtain a minimax regret solution, an algorithm based on a relaxation procedure is suggested. A numerical example demonstrates the validity and strengths of the proposed algorithm. Finally, two special cases are investigated: the minimax regret solution for fixed objective functions coefficients as well as the minimax regret solution with a reference point. Some of the characteristic features of both cases are highlighted thereafter.     

Minimax pointwise estimation of an anisotropic regression function with unknown density of the design

Our aim in this paper is to estimate with best possible accuracy an unknown multidimensional regression function at a given point where the design density is also unknown. To reach this goal, we will follow the minimax approach: it will be assumed that the regression function belongs to a known anisotropic Hölder space. In contrast to the parameters defining the Hölder space, the density of the observations is assumed to be unknown and will be treated as a nuisance parameter. New minimax rates are exhibited as well as local polynomial estimators which achieve these rates. As these estimators depend on a tuning parameter, the problem of its selection is also discussed.     

Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums

In this paper, by the generating function method, we establish various identities concerning the (higher order) Bernoulli polynomials, the (higher order) Euler polynomials, the Genocchi polynomials and the degenerate higher order Bernoulli polynomials. Particularly, some of these identities are also related to the power sums and alternate power sums. It can be found that, many well known results, especially the multiplication theorems, and some symmetric identities demonstrated recently, are special cases of our results.     

Shortened recurrence relations for Bernoulli numbers

Starting with two little-known results of Saalschütz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities.     

Spectral analysis for intrinsic time processes

This paper discusses an estimation procedure for the spectral density of intrinsic time processes because there has been no argument of the spectral analysis for subordinated processes. Such processes have been proposed in a variety of contexts to describe asset price behavior. They are used when the movement of prices is tied to the number of market transactions, trading volume or the more illusive concept of information arrival. We develop the asymptotic theory for an estimated spectral density of intrinsic time processes and elucidate the asymptotics, which show some interesting structures. Also, numerical studies are given to confirm the results.     

A local spectral approach for assessing time series model misspecification

We consider band-limited frequency-domain goodness-of-fit testing for stationary time series, without smoothing or tapering the periodogram, while taking into account the effects of parameter uncertainty (from maximum-likelihood estimation). We are principally interested in modeling short econometric time series, typically with 100 to 150 observations, for which data-driven bandwidth selection procedures for kernel-smoothed spectral density estimates are unlikely to have adequate levels. Our mathematical results take parameter uncertainty directly into account, allowing us to obtain adequate level properties at small sample sizes. The main theorems provide very general results involving joint normality for linear functionals of powers of the periodogram, while accounting for parameter uncertainty, which can be used to determine the level and power of a wide array of statistics. We discuss several applications, such as spectral peak testing and testing for the inclusion of an Unobserved Component, and illustrate our methods on a time series from the Energy Information Administration.