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Asymptotic normality of wavelet estimators of the memory parameter for linear processes. (English) Zbl 1224.62068

The authors introduce a new class M$\left(d\right)$ of real-valued processes and investigate the properties of these processes. The process $X={\left\{{X}_{k}\right\}}_{k\in ℤ}$ is an M$\left(d\right)$ process if it has the memory parameter $d$, $d\in ℝ$, and the short-range spectral density ${f}^{*}\left(\lambda \right)$. That is, for any integer $k>d-1/2$, the $k$-th order difference process ${{\Delta }}^{k}X$ is weakly stationary with spectral density function

${f}_{{{\Delta }}^{k}X}\left(\lambda \right)={|1-{e}^{-i\lambda }|}^{2\left(k-d\right)}{f}^{*}\left(\lambda \right),$

where ${f}^{*}\left(\lambda \right)$ is a non-negative symmetric function which is continuous and nonzero at the origin. The class M$\left(d\right)$ includes both stationary and nonstationary processes, depending on the value of the memory parameter $d$. The function $f\left(\lambda \right)=|1-{e}^{-i\lambda }{|}^{-2d}{f}^{*}\left(\lambda \right)$ is called the generalized spectral density of $X$. It is a proper spectral density function when $d<1/2$. In this case the process $X$ is covariance stationary with spectral density function $f\left(\lambda \right)$. The process $X$ is said to have long memory if $0, short memory if $d=0$ and negative memory if $d<0$. The process is not invertible if $d<-1/2$. The factor ${f}^{*}\left(\lambda \right)$ is a nuisance function which determines the ‘short-range’ dependence. The authors propose semi-parametric estimates for the memory parameter $d$ using wavelets from a sample ${X}_{1},\cdots ,{X}_{n}$ of the process. Two estimators of the memory parameter $d$ are considered: the log-regression wavelet estimator and the wavelet Whittle estimator. It is shown that these estimators are asymptotically normal as the sample size $n\to \infty$. An explicit expression for the limit variance is obtained. To study the asymptotic properties of these estimators the authors use a central limit theorem for scalograms, that is arrays of quadratic forms of the observed sample computed from the wavelet coefficients of this sample. For more details see F. Roueff and M.S. Taqqu [Stochastic Processes Appl. 119, No. 9, 3006–3041 (2009; Zbl 1173.60311)]. In contrast to quadratic forms computed on the basis of Fourier coefficients, such as the periodogram, the scalogram involves correlations which do not vanish as the sample size $n\to \infty$. This allows extending to the non-Gaussian linear process settings asymptotic normality results obtained for Gaussian processes. See E. Moulines, F. Roueff and M.S. Taqqu, Fractals 15, No. 4, 301–313 (2007; Zbl 1141.62073), and E. Moulines, F. Roueff and M.S. Taqqu [Ann. Stat. 36, No. 4, 1925-1956 (2008; Zbl 1142.62062).

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 62M15 Spectral analysis of processes 42C40 Wavelets and other special systems 62G05 Nonparametric estimation 60F05 Central limit and other weak theorems
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