Abstract: Strong mixing random variable sequences are used widely in practice. For example, linear processes are strongly mixing under certain conditions. In addition, some continuous time diffusion models and stochastic volatility models are strongly mixing as well. In financial risk management, population quantiles are also called VaR (Value-at-Risk) which specifies the level of excessive losses at a given confidence level. In this paper, in the presence of auxiliary information and under  strong mixing samples, the log-empirical likelihood ratio statistics for quantiles are proposed and it is shown that these statistics asymptotically have the distribution of $\chi^2$. Based on this result, the empirical likelihood based confidence intervals for quantiles are constructed. A class of testing problems are also investigated. It is shown that the asymptotic power of the testing rule in the presence of auxiliary information is higher than that without auxiliary information, and the power is not decreased as more information is available.

Key words: strong mixing sample, auxiliary information, quantile, blockwise empirical likelihood, asymptotic power