Some Applications Of Empirical Likelihood Methods

Empirical likelihood (EL), introduced by Owen (1988,1990), is a powerful non-parametric technique for constructing confidence intervals of unknown parameters. EL method has many advantages over some classic statistical inference methods, for exam-ple, EL method is more accurate than the normal approximation method in many cases especially when the underlying distribution is non-normal and the variance estimate is unstable; compare with bootstrap, EL method also has its own superiorities such as EL regions are shaped "automatically" by the sample, range preserving, transformation respecting and Bartlett correctable. Because of these advantages, EL method attracts many statisticians'attentions, and they've applied it to many different fields of statis-tics. In this thesis, we focus on developing empirical likelihood-based confidence intervals for the intermediate quantiles, conditional Value-at-Risk in ARCH/GARCH models and heteroscedastic regression models, and the ROC curve with missing data.Firstly, intermediate quantiles play an important role in the statistics of extremes with particular applications in analyzing the low-frequency-high-severity losses in finance and insurance. Therefore, developing efficient inference methodologies for intermediate quantiles is meaningful in both theory and applications. Chen & Hall (1993) proposed the so-called smoothed empirical likelihood method to construct confidence intervals for the quantiles. We further apply the EL method in Chen & Hall (1993) to construct confidence intervals for an intermediate quantile by deriving the corresponding Wilks Theorem under some extreme condition.Secondly, market risk is common seen in finance. In order to assess and control the huge loss of a financial position in financial market, VaR, i.e., Value-at-Risk is a simple and useful measure and has been widely applied in both financial institutes and risk's supervision departments. Simply speaking, VaR is defined as a threshold value such that the probability that the mark-to-market loss on the portfolio over the given time horizon exceeds this value (assuming normal markets) is the given probability level. When some volatility model is employed, conditional Value-at-Risk is of importance. As ARCH/GARCH models are widely used in modeling volatilities, we propose EL meth-ods to obtain an interval estimation for the conditional Value-at-Risk with the volatility model being an ARCH/GARCH model; we also investigate the heteroscedastic regression model, which received many attentions as it could efficiently capture the local variability in financial data. Together with local linear technique, we develop empirical likelihood confidence intervals for the conditional VaR under this model.Lastly, the receiver operating characteristic (ROC) curve is an important tool to assess the accuracy of a medical diagonstic test in discriminating diseased patients from non-diseased ones. There has been an extensive study on estimation of the ROC curve in case of complete data. However, missing data can occur in medical diagnostic studies, developing the novel methodologies for the ROC curve under this setup is valuable in real applications. We focus on the case of completely missing at random here, and by using the hot deck imputation, we propose imputation-based empirical likelihood method for the ROC curve with missing data.