Title: A Method for Estimating the Hermite rank of Long-memory Time Series

Abstract: Long-range dependent time series are often formulated by a transformation $X_t=G(Z_t)$ of an underlying long-memory Gaussian process $\{Z_t\}$. The so-called Hermite rank $m$ of $G$ critically impacts statistical inference with such a time series. Further, despite many theoretical and methodological developments for such processes over the last 50 years, no approach has existed to approximate $m$ from data. We introduce a formal method for estimating the Hermite rank $m$, which is consistent under mild conditions. The approach involves a type of sample correlation computed from squared data block averages. The method for Hermite rank estimation is examined through simulation and illustrated with two data examples, which consider log stock returns as well as tree ring widths.