Adjusted Empirical Likelihood Method and Parametric Higher Order Asymptotic Method with Applications to Finance

In recent years, applying higher order likelihood-based method to obtain inference for a scalar parameter of interest is becoming more popular in statistics because of the extreme accuracy that it can achieve. In this dissertation, we applied higher order likelihood-based method to obtain inference for the correlation coefficient of a bivariate normal distribution with known variances, and the mean parameter of a normal distribution with a known coefficient of variation. Simulation results show that the higher order method has remarkable accuracy even when the sample size is small. The empirical likelihood (EL) method extends the traditional parametric likelihood-based inference method to a nonparametric setting. The EL method has several nice properties, however, it is subject to the convex hall problem, especially when the sample size is small. In order to overcome this difficulty, Chen et al. (2008) proposed the adjusted empirical likelihood (AEL) method which adjusts the EL function by adding one ``artificial'' point created form the observed sample. In this dissertation, we extended the AEL inference to the situation with nuisance parameters. In particular, we applied the AEL method to obtain inference for the correlation coefficient. Simulation results show that the AEL method is more robust than its competitors. For the application to finance, we apply both the higher order parametric method and the AEL method to obtain inference for the Sharpe ratio. The Sharpe ratio is the prominent risk-adjusted performance measure used by practitioners. Simulation results show that the higher order parametric method performs well for data from normal distribution, but it is very sensitive to model specifications. On the other hand, the AEL method has the most robust performance under a variety of model specifications