The multi-scale high-order statistical moments of financial time series

A new high-order statistical moment based on multi-scale (MSHOM) is proposed for researching traditional statistics in this paper. In addition, the indispensable theoretical basis and derivation are illustrated in detail. With the help of three simulated time series, two kinds of situations of MSHOM analysis are mainly discussed in this work. One is accomplished by Gaussian white noise (GWN) and the other is fulfilled with Logistic map and autoregressive fractionally integrated moving-average (ARFIMA). Due to the insufficient performance of MSHOM, we propose an improved MSHOM, which is called MSHOM with control (C-MSHOM). Meanwhile, its performance is tested by the data of US and Chinese stock markets. However, C-MSHOM also brings an extra preprocess stage of data and the uncertainty of selection. To solve these problems, a more generic method, that is, generalized multi-scale high-order moments (G-MSHOM) is given in this paper.