Statistica Sinica 19 (2009), 71-81

SMOOTHED BLOCK EMPIRICAL LIKELIHOOD FOR

QUANTILES OF WEAKLY DEPENDENT PROCESSES

Song Xi Chen$^{1, 2}$ and Chiu Min Wong$^3$

$^1$Iowa State University, $^2$Peking University and $^3$Temasek Junior College

09

Abstract: Inference on quantiles associated with dependent observation is a common task in risk management. This paper employs empirical likelihood to construct confidence intervals for quantiles of the stationary distribution of a weakly dependent process. To accommodate data dependence and avoid any secondary variance estimation, empirical likelihood is formulated based on blocks of observations. To reduce the length of the confidence intervals, the weighted empirical distribution is smoothed by a kernel function. This shows that a rescaled version of the smoothed block empirical likelihood ratio admits a limiting chi-square distribution with one degree of freedom and facilitates likelihood ratio confidence intervals for quantiles. The practical performance of these confidence intervals is evaluated in a simulation study.

Key words and phrases: $alpha$-mixing, empirical likelihood, kernel smoothing, quantile, Value-at-Risk.