Some Studies On Time Series Analysis Via Generalized Empirical Likelihood Method

In modern statistical analysis,generalized empirical likelihood method attaches mounting attention due to its high-order asymptotical properties under setting a semi-parametric moment model.Compared to generalised method of moment,generalized empirical likelihood not only remains the one-order asymptotical properties,but also produces more smaller high order asymptotical bias with small sample.Additionally,as its most important members,empirical likelihood and exponential tilted likelihood are applied in a wide range of field because that they are easy to construct a confidence region or interval for unknown parameter under the chi-square approximation with no need to calculate the covariance estimates.The shape of constructed confidence region is naturally driven by data.However,all of these method are originally designed for independent data,we propose those methods to address the problems in dependent time series in this work.The details are as follows:(1)Depending on the asymptotical independence of periodograms,Monti(1997)developed frequency empirical likelihood for the statistical inference of stationary time series.In terms of computation,it still inherits the two undesirable drawbacks of its counterpart suffering from in independent data: the non-definition problem of the likelihood function and the under-coverage probability of confidence region.In chapter 2,to alleviate this dilemma,we propose two novel versions of empirical likelihood for both stationary ARMA and ARFIMA models under the assumption of Gaussian noise.Both of our proposed methods are derived by adding some pseudo samples.One is a new adjusted empirical likelihood which simultaneously eliminates the non-definition of the original empirical likelihood and enhances the coverage probability by adding two pseudo samples.Another method is mean empirical likelihood,which is constructed by adding more pseudo samples.Extensive simulations and some real data analyses indicate that our proposed methods have coverage probability more closer to nominal confidence level.(2)Exponential tilted likelihood is widely applied in many field because its likelihood ratio follows a chi-square limiting distribution,which makes it convenient for interval estimation and its point estimate is more robust than empirical likelihood if the moment models are mis-specified.Similar to empirical likelihood,an optimization is involved in the computation of this kind of semi-parametric likelihood supported on data under certain convex constrains.If these constrains are nonlinear,the computation will loss its advantage.In chapter 3,we propose not only exponential tilted but also jackknife exponential tilted method for constructing confidence region for stationary linear model borrowing the idea of Jing & Yuan(2009).By applying the jackknife method to Whittle estimator,we obtain new asymptotically independent pseudo samples which will be used to construct linear constrains for likelihood ratio.The proposed exponential tilted likelihood ratio and jackknife exponential tilted likelihood ratio are both shown to follow a chi-square limiting distribution.But these methods are both suffering from the non-definition and large coverage error.To overcome these problems,we further proposed the adjusted method with adjustment level proposed by Chen et al.(2008).Our simulation studies indicate that the adjusted technique does achieve a higher-order coverage precision than the unadjusted exponential tilted method.In addition,due to the good performance of exponential tilted point estimation under over-identified moment model misspecification,we apply the result to the autoregressive coefficient of AR(1)by deriving the misspecified estimating equations from spectral density.And the one-order property of point estimate is preserved.Simulation results verify that the point estimates of the exponential tilted outperform those of empirical likelihood.(3)For estimating multiple change-points in piecewise stationary processes,model selection is an important approach.However,the model selection,involving the optimization of some penalty loss function,makes the computation extremely expensive because the number of possible change point combinations growing exponentially with the increasing length of data sequence.In chapter 4,we first construct two-sample empirical likelihood method based on the independent sample under conditional likelihood.The likelihood ratio scan statistic is used to detect the number and position of potential change points.Then following the idea of Davis & Yau(2013),we replace the conditional likelihood function by empirical likelihood in their objective function of minimum discription length to estimate the change points,which makes the assumption of model more flexible by the non-parametric nature of empirical likelihood.This step remains the computational complexity proposed by Yau & Zhao(2016)under the obtained initial change estimates from the first step.Finally,we obtain the optimal change estimates by using the empirical likelihood ratio of Baragona et al.(2013).Simulation studies verify that our proposed method is superior to estimate the number and locations of change points for small sample.