Empirical Likelihood Inference For Expectile

Since expectile was proposed by Aigner, Amemiya and Poirier in1976, it has been rapidly developped and wide applied. For example, in financial terms, a varying-coefficient expectile model for estimating value at risk was proposed by Alan T.K.Wan in 2013. When the extreme loss occurs in market, expectile can be well used to measure the size of extreme loss, thereby achieve the purpose of prediction and prevention.In the past, the research in expectile focused on the estimated value of a single expectile, or estimate a smooth curve of expectile function.The main methods include spline interpolation method, normal approximation method, asymmetric least squares method(ALSM),bootstrap method, etc. Using spline interpolation method to constructed point estimation of expectile function, having a shortcomings of low accuracy. Using the normal approximation to estimate the confidence interval estimation about expectile, large sample dependent on the limit of variance, the effect is not good for small sample.Empirical likelihood method as a non-parametric method of inference has many excellent statistical properties. When the sample size tends to infinity, the empirical likelihood ratio statistics converges to 1-degree of freedom for the chi-square distribution, using the limiting distribution to construction the confidence region about parameter doesnot depend on the structure of the limit about variance, and the shape of confidence regions only depends on the sample data, the constructed confidence intervals also has a region retention and translation invariance.These excellent properties about empirical likelihood make some remedy on spline interpolation method and the normal approximation method, therefore, the main work of this paper is to use the empirical likelihood method for statistical inference of expectile.The thesis consists of five chapters. Chapter 1, an introduction, the origin of expectile, research process and application status about empirical likelihood and expectile. Chapter 2, preliminaries, introduces the concept of expectile, the relation between empirical likelihood and estimating equation. Chapter 3, with the empirical likelihood method to estimate expectile, construct the empirical log-likelihood ratio statistic of expectile, prove empirical log-likelihood ratio statistic of expectile is asymptotic subject to the chi-square distribution with 1-degree of freedom, make use of the asymptotic distribution, constructed the confidence regions of expectile. Chapter 4, simulation studies, the samples form randomly generated in different distributions, consider the coverage of these samples at different confidence levels corresponding to different weights about expectile. The simulation results show that the empirical likelihood method to estimate the coverage about expectile close to the desired level, results better than the normal approximationmethod. Chapter 5, summary and prospect.