Asymptotics of rank order statistics for ARCH residual empirical processes

This paper gives the asymptotic theory of a class of rankorder statistics (T_N) for two-sample problem pertaining to empirical processes based on the squared residuals from two classes of ARCH models. An important aspect is that, unlike the residuals of ARMA models, the asymptotics of (T_N) depend on those of ARCH volatility estimators. Such asymptotics provide a useful guide to the reliability of confidence intervals, asymptotic relative efficiency and ARCH affection. We consider these aspects of (T_N) for some ARCH residual distributions via numerical illustrations. Moreover, a measure of robustness for (T_N) is introduced. These studies help to highlight some important features of ARCH residuals in comparison with the i.i.d. or ARMA settings.