Long Memory Errors Research Articles

Wavelet estimation for the nonparametric additive model in random design and long-memory dependent errors

We investigate the nonparametric additive regression estimation in random design and long-memory errors and construct adaptive thresholding estimators based on wavelet series. The proposed approach achieves asymptotically near-optimal convergence rates when the unknown function and its univariate additive components belong to Besov space. We consider the problem under two noise structures; (1) homoskedastic Gaussian long memory errors and (2) heteroskedastic Gaussian long memory errors. In the homoskedastic long-memory error case, the estimator is completely adaptive with respect to the long-memory parameter. In the heteroskedastic long-memory case, the estimator may not be adaptive with respect to the long-memory parameter unless the heteroskedasticity is of polynomial form. In either case, the convergence rates depend on the long-memory parameter only when long-memory is strong enough, otherwise, the rates are identical to those under i.i.d. errors. In addition, convergence rates are free from the curse of dimensionality.

Asymptotic properties of wavelet estimators in partially linear errors-in-variables models with long-memory errors

While the random errors are a function of Gaussian random variables that are stationary and long dependent, we investigate a partially linear errors-in-variables (EV) model by the wavelet method. Under general conditions, we obtain asymptotic representation of the parametric estimator, and asymptotic distributions and weak convergence rates of the parametric and nonparametric estimators. At last, the validity of the wavelet method is illuminated by a simulation example and a real example.

Multichannel deconvolution with long range dependence: Upper bounds on the [formula omitted]-risk [formula omitted

We consider multichannel deconvolution in a periodic setting with long-memory errors under three different scenarios for the convolution operators, i.e., super-smooth, regular-smooth and box-car convolutions. We investigate global performances of linear and hard-thresholded non-linear wavelet estimators for functions over a wide range of Besov spaces and for a variety of loss functions defining the risk. In particular, we obtain upper bounds on convergence rates using the Lp-risk (1≤p<∞). Contrary to the case where the errors follow independent Brownian motions, it is demonstrated that multichannel deconvolution with errors that follow independent fractional Brownian motions with different Hurst parameters results in a much more involved situation. An extensive finite-sample numerical study is performed to supplement the theoretical findings.

On estimation of mean and covariance functions in repeated time series with long-memory errors*

We consider kernel estimation of trend and covariance functions in models typically encountered in functional data analysis (FDA), with the modification that the random curves are perturbed by error processes that exhibit short- or long-range dependence. Uniform convergence of standardized maximal differences between estimated and true (trend and covariance) functions is established. For the covariance function, a transformation based on contrasts is proposed that does not require explicit trend estimation. Improved estimators can be obtained by using higher-order kernels.

RETRACTED: Regression estimation with locally stationary long-memory errors

This article has been retracted: please see Elsevier Policy on Article Withdrawal ( http://www.elsevier.com/locate/withdrawalpolicy ). This article has been retracted at the request of the Editor-in-Chief, who noted an administrative error. The EiC accidentally accepted while the reviewers’ recommendation was to reject.We apologize that this was not detected during the submission and review process.

Asymptotic expansion for nonparametric M-estimator in a nonlinear regression model with long-memory errors

We consider asymptotic expansion of the nonparametric M-estimator in a fixed-design nonlinear regression model when the errors are generated by long-memory linear processes. Under mild conditions, we show that the nonparametric M-estimator is first-order equivalent to the Nadaraya–Watson (NW) estimator, which implies that the nonparametric M-estimator has the same asymptotic distribution as that of the NW estimator. Furthermore, we study the second-order asymptotic expansion of the nonparametric M-estimator and show that the difference between the nonparametric M-estimator and the NW estimator has a limiting distribution after suitable standardization. The nature of the limiting distribution depends on the range of long-memory parameter α . We also compare the finite sample behavior of the two estimators through a numerical example when the errors are long-memory.

Edgeworth Expansion for Linear Regression Processes with Long-Memory Errors

This article provides an Edgeworth expansion for the distribution of the log-likelihood derivative LLD of the parameter of a time series generated by a linear regression model with Gaussian, stationary, and long-memory errors. Under some sets of conditions on the regression coefficients, the spectral density function, and the parameter values, an Edgeworth expansion of the density as well as the distribution function of a vector of centered and normalized derivatives of the plug-in log-likelihood PLL function of arbitrarily large order is established. This is done by extending the results of Lieberman et al. (2003), who provided an Edgeworth expansion for the Gaussian stationary long-memory case, to our present model, which is a linear regression process with stationary Gaussian long-memory errors.

A Note on Block-Thresholded Wavelet Estimators with Correlated Noise

Hall et al. (1999) proposed block-thresholding methods to estimate mean regression functions with independent random errors. They showed that block-thresholded wavelet estimators attain minimax-optimal convergence rates when the mean functions belong to a large class of functions that involve a wide variety of irregularities, including chirp and Doppler functions, and functions with jump discontinuities. In this article, we show that block-thresholded wavelet estimators still attain minimax convergence rates when the mean functions belong to a wide range of Besov classes (where s > 1/p, p ≥ 1 and q ≥ 1) with long-memory Gaussian errors. Therefore, in the presence of long-memory Gaussian errors, wavelet estimators still provide extensive adaptivity.

Signal Sampling and Recovery Under Dependent Errors

The paper examines the impact of the additive correlated noise on the accuracy of the signal reconstruction algorithm originating from the Whittaker-Shannon (WS) sampling interpolation formula. A class of band-limited signals as well as signals which are non-band-limited are taken into consideration. The proposed reconstruction method is a smooth post-filtering correction of the classical WS interpolation series. We assess both the point-wise and global accuracy of the proposed reconstruction algorithm for a broad class of dependent noise processes. This includes short and long-memory stationary errors being independent of the sampling rate. We also examine a class of noise processes for which the correlation function depends on the sampling rate. Whereas the short-memory errors have relatively small influence on the reconstruction accuracy, the long-memory errors can greatly slow down the convergence rate. In the case of the noise model depending on the sampling rate further degradation of the algorithm accuracy is observed. We give quantitative explanations of these phenomena by deriving rates at which the reconstruction error tends to zero. We argue that the obtained rates are close to be optimal. In fact, in a number of special cases they agree with known optimal min-max rates. The problem of the limit distribution of the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> distance of the proposed reconstruction algorithm is also addressed. This result allows us to tackle an important problem of designing non- parametric lack-of-fit tests. The theory of the asymptotic behavior of quadratic forms of stationary sequences is utilized in this case.

On the asymptotic variance in nonparametric regression with fractional time-series errors

This paper focuses on deriving explicit formulae of the asymptotic variance in nonparametric regression with fractional time-series errors. Unified formulae are obtained for fixed design models with long-memory, short-memory and antipersistent errors. It is also found that in strongly antipersistent case the Uniform kernel is no longer the minimum variance one and that for a fourth-order kernel the constant in the asymptotic variance for long-memory errors may be clearly smaller than that for i.i.d. errors. The results are applied to improving an existing data-driven algorithm. Practical performance of the proposed algorithm is illustrated with simulated and real data examples.

Polynomial Trend Regression With Long-memory Errors

Abstract. For a time series generated by polynomial trend with stationary long-memory errors, the ordinary least squares estimator (OLSE) of the trend coefficients is asymptotically normal, provided the error process is linear. The asymptotic distribution may no longer be normal, if the error is in the form of a long-memory linear process passing through certain nonlinear transformations. However, one hardly has sufficient information about the transformation to determine which type of limiting distribution the OLSE converges to and to apply the correct distribution so as to construct valid confidence intervals for the coefficients based on the OLSE. The present paper proposes a modified least squares estimator to bypass this drawback. It is shown that the asymptotic normality can be assured for the modified estimator with mild trade-off of efficiency even when the error is nonlinear and the original limit for the OLSE is non-normal. The estimator performs fairly well when applied to various simulated series and two temperature data sets concerning global warming.

Asymptotic inference for LSE in multivariate continuous regression models with long-memory random fields

This paper establishes the asymptotic behaviour for the covariance matrix and the limit distributions of the least squares estimators for a regression coefficients in a multivariate continuous regression models with long-memory Gaussian errors. The used method is based on the asymptotic analysis of orthogonal expansion of non-linear functionals of homogeneous and isotropic Gaussian random fields.

On the rate of convergence to the normal law for LSE in multivariate continuous regression model with long-range dependence stationary errors

In this paper, we focus on a multivariate continuous regression model with long-memory stationary Gaussian errors. Some upper bounds on the rate of convergence in the Central Limit Theorem for normalized least square estimators (LSE) in these regression models are obtained. The used method is based on the asymptotic analysis of orthogonal expansion of non linear functionals of stationary Gaussian processes and on the Kolmogorov distance.

Nonparametric M-estimation with long-memory errors

We investigate the behavior of nonparametric kernel M-estimators in the presence of long-memory errors. The optimal bandwidth and a central limit theorem are obtained. It turns out that in the Gaussian case all kernel M-estimators have the same limiting normal distribution. The motivation behind this study is illustrated with an example.

Local polynomial fitting with long-memory, short-memory and antipersistent errors.

Local polynomial smoothing for the trend function and its derivatives in nonparametric regression with long-memory, short-memory and antipersistent errors is considered. We show that in the case of antipersistence, the convergence rate of a nonparametric regression estimator is faster than for uncorrelated or short-range dependent errors. Moreover, it is shown that unified asymptotic formulas for the optimal bandwidth and the MSE hold for all of the three dependence structures. Also, results on estimation at the boundary are included. A bandwidth selector for nonparametric regression with different types of dependent errors is proposed. Its asymptotic property is investigated. The practical performance of the proposal is illustrated by simulated and real data examples.

Wavelet-Generalized Least Squares: A New BLU Estimator of Linear Regression Models with 1/f Errors

Long-memory noise is common to many areas of signal processing and can seriously confound estimation of linear regression model parameters and their standard errors. Classical autoregressive moving average (ARMA) methods can adequately address the problem of linear time invariant, short-memory errors but may be inefficient and/or insufficient to secure type 1 error control in the context of fractal or scale invariant noise with a more slowly decaying autocorrelation function. Here we introduce a novel method, called wavelet-generalized least squares (WLS), which is (to a good approximation) the best linear unbiased (BLU) estimator of regression model parameters in the context of long-memory errors. The method also provides maximum likelihood (ML) estimates of the Hurst exponent (which can be readily translated to the fractal dimension or spectral exponent) characterizing the correlational structure of the errors, and the error variance. The algorithm exploits the whitening or Karhunen–Loéve-type property of the discrete wavelet transform to diagonalize the covariance matrix of the errors generated by an iterative fitting procedure after both data and design matrix have been transformed to the wavelet domain. Properties of this estimator, including its Cramèr–Rao bounds, are derived theoretically and compared to its empirical performance on a range of simulated data. Compared to ordinary least squares and ARMA-based estimators, WLS is shown to be more efficient and to give excellent type 1 error control. The method is also applied to some real (neurophysiological) data acquired by functional magnetic resonance imaging (fMRI) of the human brain. We conclude that wavelet-generalized least squares may be a generally useful estimator of regression models in data complicated by long-memory or fractal noise.

Finite sample efficiency of OLS in linear regression models with long-memory disturbances

OLS is as efficient as GLS in the linear regression model with long-memory errors as the long-memory parameter approaches the boundary of the stationarity region, provided the model contains a constant term. This generalizes previous results of Samarov and Taqqu [Journal of Time Series Analysis 9 (1988) 191–200] and gives a further example of the ‘high-correlation asymptotics’ of Krämer and Baltagi [Economics Letters 50 (1996) 13–17].

Second-order behavior of M-estimators in linear regression with long-memory errors

This paper obtains a higher-order asymptotic expansion of a class of M-estimators of the regression parameter in a one-parameter linear regression model when the errors form a long-memory moving average. Suitably standardized difference between an M-estimator and the least-squares estimator is shown to have a limiting distribution. The nature of the limiting distribution depends on the range of the dependence parameter θ . For example, in the case of the symmetric common error distribution, if 1/3< θ <1, then a suitably standardized difference between the least absolute deviation and the least-squares estimators converges weakly to a normal distribution. If 0< θ <1/3 then the corresponding limiting distribution is not normal.

How to overcome the curse of long-memory errors

Long-memory errors dramatically slow down the convergence of minimax risks in a fixed design nonparametric regression. The problem becomes even more complicated for the case of adaptive estimation. This defines the curse of long-memory errors. I show that using a random design, instead of a fixed one, allows one to overcome this curse and make familiar data-driven estimators robust. Moreover, the result holds for a wide class of nonstationary errors with bounded moments (including bounded deterministic errors). Possible extensions are discussed.

On preliminary test and shrinkage estimation in linear models with long-memory errors

This paper discusses the problem of estimating a subset of parameters when the complementary subset is possibly redundant, in a linear regression model when the errors are generated from a long-memory process. Such a model arises due to the overmodelling of a situation involving long-memory data. Along with the classical least-squares estimator and restricted least-squares estimator, preliminary test least-squares estimator and shrinkage least-squares estimator are investigated in an asymptotic set-up and their relative performances are studied under contiguous alternatives. The contiguous alternatives under such dependence are fundamentally different from those under the independent errors case.