The mode of the climacogram estimator for a Gaussian Hurst-Kolmogorov process

Geophysical processes are often characterized by long-term persistence. An important characteristic of such behaviour is the induced large statistical bias, i.e. the deviation of a statistical characteristic from its theoretical value. Here, we examine the most probable value (i.e. mode) of the estimator of variance to adjust the model for statistical bias. Particularly, we conduct an extensive Monte Carlo analysis based on the climacogram (i.e. variance of the average process vs. scale) of the simple scaling (Gaussian Hurst-Kolmogorov) process, and we show that its classical estimator is highly skewed especially in large scales. We observe that the mode of the climacogram estimator can be well approximated by its lower quartile (25% quantile). To derive an easy-to-fit empirical expression for the mode, we assume that the climacogram estimator follows a gamma distribution, an assumption strictly valid for Gaussian white noise processes. The results suggest that when a single timeseries is available, it is advantageous to estimate the Hurst parameter using the mode estimator rather than the expected one. Finally, it is discussed that while the proposed model for mode bias works well for Gaussian processes, for higher accuracy and non-Gaussian processes, one should perform a Monte Carlo simulation following an explicit generation algorithm.