一种改进的有色噪声背景下信号的检测算法

摘要:对于自回归型(AR)的广义平稳(WSS)高斯有色噪声,以改进的协方差法得到了该噪声未知参数基于AR模型的估计,该估计优于现在广泛使用的周期图方法.然后利用渐进似然函数,建立了广义似然比(GLRT)检测器,该检测器的性能依赖于噪声的未知参数估计的精度,并且得到了检测概率和虚警概率的渐进表达式.通过对含有WSS高斯有色噪声的信号在不同信噪比下的计算机仿真实验分析,结果验证了提出方法的有效性,而且检测性能明显优于原算法.     

改进周期图法功率谱估计中的窗函数仿真分析

周期图法是功率谱估计的一种基本方法,但该方法不满足一致性估计条件,谱估计的分辨力和方差都很难满足实际应用需要.因此采用其改进方法(即Welch法)估计信号功率谱,并结合该方法对四种典型窗函数的谱估计性能进行了具体的仿真,就其中的频率分辨率、采样信号的数据长度与窗函数的特性等因素之间的关系,进行了详细的讨论,重点分析了窗函数对谱估计的影响,指出了它们的优缺点,得出了不同数据类型与估计方式下的功率谱标准方差的均值.最后,根据实验结果并结合窗函数的评价指标,指出了选取合适的窗函数应注意的一般原则,具有一定的实际指导意义.     

The mean squared error of Geweke and Porter-Hudak's estimator of the memory parameter of a long-memory time series

We establish some asymptotic properties of a log-periodogram regression estimator for the memory parameter of a long-memory time series. We consider the estimator originally proposed by Geweke and Porter-Hudak (The estimation and application of long memory time series models. Journal of Time Ser. Anal. 4 (1983), 221–37). In particular, we do not omit any of the low frequency periodogram ordinates from the regression. We derive expressions for the estimator's asymptotic bias, variance and mean squared error as functions of the number of periodogram ordinates, m , used in the regression. Consistency of the estimator is obtained as long as m ←∞ and n ←∞ with ( m log m )/ n ← 0, where n is the sample size. Under these and the additional conditions assumed in this paper, the optimal m , minimizing the mean squared error, is of order O( n ^4/5). We also establish the asymptotic normality of the estimator. In a simulation study, we assess the accuracy of our asymptotic theory on mean squared error for finite sample sizes. One finding is that the choice m = n ^1/2, originally suggested by Geweke and Porter-Hudak (1983), can lead to performance which is markedly inferior to that of the optimal choice, even in reasonably small samples.     

Plug-in Selection of the Number of Frequencies in Regression Estimates of the Memory Parameter of a Long-memory Time Series

We consider the problem of selecting the number of frequencies, m , in a log-periodogram regression estimator of the memory parameter d of a Gaussian long-memory time series. It is known that under certain conditions the optimal m , minimizing the mean squared error of the corresponding estimator of d , is given by m ^(opt)= Cn ^4/5, where n is the sample size and C is a constant. In practice, C would be unknown since it depends on the properties of the spectral density near zero frequency. In this paper, we propose an estimator of C based again on a log-periodogram regression and derive its consistency. We also derive an asymptotically valid confidence interval for d when the number of frequencies used in the regression is deterministic and proportional to n ^4/5. In this case, squared bias cannot be neglected since it is of the same order as the variance. In a Monte Carlo study, we examine the performance of the plug-in estimator of d , in which m is obtained by using the estimator of C in the formula for m ^(opt) above. We also study the performance of a bias-corrected version of the plug-in estimator of d . Comparisons with the choice m = n ^1/2 frequencies, as originally suggested by Geweke and Porter-Hudak (The estimation and application of long memory time series models. Journal of Time Ser. Anal. 4 (1983), 221–37), are provided.     

TESTING EQUALITY OF VARIANCES FOR PAIRED TIME SERIES

Abstract. We develop a simple test for testing equality of variances for paired stationary Gaussian time series. The test statistic is a modified z statistic. It is based on the periodograms of the two series and consistent estimation of the difference between the two spectral densities. Simulations illustrate the validity of the asymptotic results for finite samples. An application to EEG data is discussed.     

An Unbiased Measure of Integrated Volatility in the Frequency Domain

This article studies the effect of market microstructure noise on volatility estimation in the frequency domain. We propose a bias‐corrected periodogram‐based estimator of integrated volatility. We show that the new estimator is consistent and the central limit theorem is established under a general assumption of the noise. We also provide a feasible procedure for computing the bias‐corrected estimator in practice. As a byproduct, we extract a consistent frequency‐domain estimator of the long‐run variance of market microstructure noise from high‐frequency data.     

Sequential Change-Point Detection Methods for Nonstationary Time Series

We present two new spectral-based methods for detection of changes in autocorrelation structure in a continuous-valued time series in an online process monitoring setting. Our methods are based on the idea that changes in the autocorrelation structure are reflected by changes in the Fourier or wavelet-based spectrum and can be detected by comparing estimated spectra of adjacent blocks of the series. To be effective for slowly changing spectral structure, the methods are extended to allow information from more than one past block to be used in determining whether a change has occurred, in such a way as to minimize computational burden. Through simulation, we evaluate the performance of our methods and find that they can provide reliable and timely detection of changes in covariance structure in an online monitoring framework. We illustrate the methods using electroencephalogram traces (brain waves) and run-time computer performance metrics.     

An Efficient Taper for Potentially Overdifferenced Long-memory Time Series

We propose a new complex-valued taper and derive the properties of a tapered Gaussian semiparametric estimator of the long-memory parameter d ε (−0.5, 1.5). The estimator and its accompanying theory can be applied to generalized unit root testing. In the proposed method, the data are differenced once before the taper is applied. This guarantees that the tapered estimator is invariant with respect to deterministic linear trends in the original series. Any detrimental leakage effects due to the potential noninvertibility of the differenced series are strongly mitigated by the taper. The proposed estimator is shown to be more efficient than existing invariant tapered estimators. Invariance to k th order polynomial trends can be attained by differencing the data k times and then applying a stronger taper, which is given by the k th power of the proposed taper. We show that this new family of tapers enjoys strong efficiency gains over comparable existing tapers. Analysis of both simulated and actual data highlights potential advantages of the tapered estimator of d compared with the nontapered estimator.     

Simultaneous estimation of the input and output frequencies of nonlinear systems

In this paper, the simultaneous estimation of the input and output frequencies of nonlinear systems is considered. As the output frequencies are generated from the input frequencies, and are integer combinations of these frequencies, it is shown in this paper that the simultaneous estimation of both the input and output frequencies can therefore be formulated as a constrained estimation problem. First, the constrained Cramér-Rao lower bound, an important general property of any unbiased estimator, is derived. The procedure and algorithm for estimating the input and output frequencies are devised based on the periodogram method. Numerical examples are presented to illustrate the performance and implementation of the proposed estimation procedure and algorithm.     

EXACT MAXIMUM LIKELIHOOD ESTIMATE AND LAGRANGE MULTIPLIER TEST STATISTIC FOR ARMA MODELS

Abstract. The paper provides a method for the computation of the derivatives of the exact log likelihood function of a Gaussian time series. Based on this result and using Fisher's scoring technique, an efficient method for computing the maximum likelihood estimates for an autoregressive moving average model has been obtained. Simulations suggest that the new procedure is as fast as the Box and Jenkins conditional least squares method. In a similar way, a procedure is derived to compute the Lagrange multiplier test statistics for testing the goodness of fit of the model.