Testing the Tail Index in AR Model and Average AR Quantile
Jana Jureckova, Charles University and The Czech Academy of Sciences, Czech Republic
The talk is partially based on the joint results with Hira L. Koul. We consider the linear autoregressive model of order p with possibly heavy tailed innovations. The AR quantiles and the dual AR rank scores were introduced by Koul and Saleh in 1995 following the regression quantiles and rank scores of the linear regression model. Specifically, the averaged AR alpha-quantile follows the tail behavior of the innovations, it is monotone in alpha, and it enables inference on distribution properties of the innovations. Being solution to a specific linear programming problem, under every number of observations, it is a linear combination of a finite number (p) of observations corresponding to the optimal base. The extreme averaged AR quantile is of a particular interest; besides being a linear combination of p basic observations, it can be expressed by means of (rank) R-estimate of the AR coefficients. A frequent problem of interest is test of hypothesis on the tail index of the innovations. An asymptotic nonparametric test, based on the empirical process of maxima of segments of the time series, was constructed by Koul and coauthors in 2009. Another test can be likely based also on the averaged AR quantiles.