Robust estimation of the sample mean variance for Gaussian processes with long-range dependence

We introduce a method to robustly estimate the variance of the sample mean for two Gaussian processes exhibiting Long-Range Dependence (LRD): fractional Gaussian noise, fGn(H), and fractional autoregressive integrated moving average, FARIMA(0, H, 0) processes. Our method is based on a combination of two estimators of H which are functions of a multitaper estimator of the log spectral density: a robust estimator which uses a limited number of frequencies near the origin, and an estimator which uses (almost) all of the frequencies. An important feature of our method is a test to differentiate between fGn(H) and FARIMA(0, H, 0) which, under a correct decision, allows us to estimate the Hurst parameter H using the correct model specification. Differentiating between fGn(H) and FARIMA(0, H, 0) with Gaussian innovations is important to estimate the sample mean variance and for statistical inference about the process location parameter. Numerical comparisons are made against existing estimators including parametric and semi-parametric methods in Fourier frequency and wavelet domains.