The Memory Parameter Study In Long-memory Time Series

Long memory of time series is first presented in 1951 by Hurst, who is a hydrolo-gist. And then Mandelbrot introduced the concept of fractional brown motion and fractal, which laid a rigorous mathematical foundation of long memory analysis. Long memory was widely attracted attention in fluid science, meteorology, geophysics and other natural sci-ences. Later, many scholars at home and abroad take a great deal of empirical analysis, which used auto-correlation coefficient, KPASS test, GKL testing, rescaled range analysis (R/S), the amendments of the R/S (MRS) and other methods to test the long memory of the sequence of the financial market return and fluctuations in the rate of sequence. For example, in the stock markets of United States, India, Britain, China and other countries, it is concluded that the stock market has the property of long memory. It questioned the theory of efficient markets. Therefore, the time-series models (ARMA, ARIMA, ARCH, GARCH model, etc.), which is used to describe the short memory, will no longer be suitable. We must think about the factor of long memory.Now, the long memory of stationary time series is described by the function of correla-tion coefficient:ρ(?)((?)is used for the lag coefficient).ρ(?)≈C(?)2d-1,当(?)→∞, d∈(0,1/2), Cis nonzero We know that, for the function of d∈(0,1/2), its self-correlation coefficient is positive. It delayed by curve. Its sum goes to infinity. (It also can be defined by the spectral density function). When d get the other values, time series does not have long memory. Therefore, in order to study the long-memory of time series, it is important to estimate the fractal differ-ential parameters:d (It is also called memory parameter). d expressed long-term dependence of time series. We will study the memory parameter in this paper. The organizational struc-ture of this article:first, we will introduce the concept of the long memory of time series and the memory parameter. Then, we study the relationship between usual parameters in the long memory time series and memory parameter. Second, we summarize the estimation of memory parameter, when time series are normal distribution, there are two methods: parametric and semi-parametric. Analyze the pros and cons of estimation methods. Third, when time series are not normal distribution, we suppose that time series has the form of heavy-tails which is one of the hot issue many people study. With the conclusion have done, we will give a new method which is based on semi-parametric estimation method.Because heavy-tailed distribution of time series is one kind and it dose not have a specific form of distribution function, we only study the time series subject to a specific case of heavy-tailed distribution, it is also called symmetric a-stable distribution. At this point, time series is called symmetric a stable random process. Assume that the random variable subject to symmetricα-stable distribution, it is certainly heavy-tailed. So far, many people study memory parameter estimation in this case. We will obtain a variety of estimation methods above all. And then give a new memory parameter estimation method and it's properties.