Graphical Methods for Investigating the Finite-sample Properties of Confidence Regions: an application to long memory

In the literature, there are not satisfactory methods for measuring and presenting the performance of confidence regions. In this paper, techniques for measuring effectiveness of confidence regions and for the graphical display of simulation evidence concerning the coverage and effectiveness of confidence regions are developed and illustrated. Three types of figures are discussed: called coverage plots, coverage discrepancy plots, and coverage effectiveness curves, that permits to show the ``true'' effectiveness, rather than a spurious nominal effectiveness. We demonstrate that when simulations are run to compute the coverage for only one confidence level, which is done for classical presentations in tables, all the information useful for building the coverage plot is present. Thus, there is absolutely no loss of computing time by using this method. These figures are used to illustrate the finite sample properties of long range dependence confidence regions. Particularly, we present and comment classical confidence intervals and confidence intervals based on inverting bootstrap tests for the long range dependence parameter in the ARFIMA models. Monte Carlo results on these confidence intervals for various situations are also presented. We show that classical confidence intervals have very poor performances, even the percentile-t interval, whereas confidence intervals based on inverting bootstrap tests have quite satisfactory performance. These intervals are then applied on the S&P500 index to illustrate a realistic case