In this paper, it is proposed to estimate the memory parameter of a potentially long-range dependent time series by applying goodness-of-fit tests to the cumulative normalized periodogram in the neighborhood of frequency zero. The results of an extensive simulation study show that this new estimator performs well compared to conventional frequency-domain estimators which are based on the Whittle likelihood or are obtained from the popular log periodogram estimator by trimming, smoothing, and utilizing non-Fourier frequencies, respectively. In an empirical investigation of log absolute daily index returns, we find evidence of long-range dependence with values of the memory parameter in the range between 0.2 and 0.3 both in developed and developing stock markets. There are no indications of long-range dependence in the case of the original index returns.