Statistical Inference Of Tail Index In Extreme Value Theory

In extreme value theory,the so-called tail index is a parameter that describes the heavy-tailed behavior of a distribution,so understanding this parameter is crucial to solving many of the problems associated with extreme events.In this thesis,we apply empirical likelihood and local polynomial methods to the statis-tical inference of the tail index of Pareto-type distribution when the covariates are random,and further prove the asymptotic properties of the estimator including consistency and asymptotic normality.This thesis includes several components as follows:First,we consider the estimation of the confidence region for the regression coefficient in the tail index regression model when the covariates are random.In that case,we adopt the methodology of Owen(1991)to prove that the log empirical likelihood ratio function is asymptotic ?2(p)distributed.In most cases,the results of simulation studies indicate that the empirical likelihood is better than the normal approximation(Wang and Tsai,2009)in terms of coverage probability.Especially in the actual case,the empirical likelihood method gives a narrower interval than the normal approximation method.Second,in the past three decades nonparametric regression or smoothing has had a huge impact in many areas of applied statistics.However,its application to extreme value data analysis has been rather slow and fragmented.On this basis,we further consider how to combine varying coefficient method with extreme value theory in order to obtain efficient estimator of the tail index.In this thesis,we will generalize the parametric tail index regression model,proposed by Wang and Tsai(2009)as estimator for the conditional tail index in random covariate case,to the varying coefficient setting.By using the local linear fitting method of Fan and Gijbels(1996),we obtain an approximate local maximum likelihood estimator and prove its asymptotic normality.The results of simulation studies show that the estimated coefficient functions and constant coefficients are close to the true value even when the dimension of covariates is high.Finally,we study the nonparametric estimation of the conditional tail in-dex for heavy-tailed distributions in the random covariate case.Similarly,by applying the local polynomial method of Fan and Gijbels(1996),we obtain an approximate local maximum likelihood estimator and prove its consistency and asymptotic normality.The numerical results show that,at least for the Burr distribution we give,the proposed nonparametric tail index regression(NTIR)estimator produces more robust estimate of the conditional tail index than the Goegebeur et al.(2014a),Goegebeur et al.(2014b)and Beirlant and Goegebeur(2004).In addition,we apply the proposed model to a practical example,and give the estimated tail index and its 95%pointwise confidence intervals with bias ignored.