Research On Spectrum Analysis Of Parameter Model Based On Stable Process And Field

Frequency domain analysis, also known as spectral analysis, mainly researchsignals of various features in the frequency domain. The estimation and analysisof power spectrum are important in many engineering applications. This paperdiscusses several probability performances and index evaluation of heavy-tailedstable distribution, which is an important part of the distribution having regularlyvarying. At first, the probability and statistics features of the α stable distributionare introduced as the theoretical basis of this study. Secondly, sets forth fourrepresentations of stable random variables characteristic function, clears themeaning of each parameter and finds relations between parameters in theserepresentations. This paper prove that these properties studied in standardparameterization are equally applicable to other parameterizations,Then we build Hubor’s М-estimation for autoregressive process withsymmetric stable residuals. On the basis of proving autoregressive processes withresiduals, which are Alpha stable random variables, are also stationary processes,and using Markov’s inequality and related theories to discuss Hubor’sМ-estimation for autoregressive process with symmetric stable residuals, weprove the consistency and asymptotic normality of this estimate further, whichresolve successfully autoregressive process with symmetric stable residualsproblem.Finally, this paper studies parameters estimation and heavy tail behavior ofGARCH(1,1) with the errors having regularly varying distribution, and shows theheavy tail behavior of GARCH (1,1) process with Alpha-stable residuals {t}t∈Z,α∈(0,2] and {t}t∈Zerrors. And then modify maximum likelihood function andprove the credibility and asymptotically normal distribution of the estimationwith related theories.This paper solves the following problems: clears the meaning of each parameter and finds relations between parameters in these representations; provesthe consistency and asymptotic normality of this Hubor’s М-estimation forautoregressive process with symmetric stable residuals; builds parametersestimation and heavy tail behavior of GARCH (1,1) with the errors havingregularly varying distribution, and obtains index estimate of stable heavy-taileddistribution.