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We propose and compare several methods of constructing wavelet-based confidence intervals for the self-similarity parameter in heavy-tailed observations. We use empirical coverage probabilities to assess the procedures by applying them to Linear Fractional Stable Motion with many choices of parameters. We find that the asymptotic confidence intervals provide empirical coverage often much lower than nominal. We recommend the use of resampling confidence intervals. We also propose a procedure for monitoring the constancy of the self-similarity parameter and apply it to Ethernet data sets.