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## 4.3 Change in the tail index 

To test for a change in the tail index, we choose ψ ( x ) = log( x $^{2}$). 

CUSUM: In this case, E( ψ ( X$_{1}$ ) | F $_{0}$) ̸ = 0 and hence long memory affects the limiting distribution of the CUSUM statistic. Moreover, 

<img src='content_image/316996.jpg'>

so that the Hermite rank of Ψ equals the Hermite rank of h = log ◦ σ . 

We note further that if ψ ( x ) = log( x $^{2}$), we have 

<img src='content_image/316997.jpg'>

The first summand converges to 2( J$_{m}$ (log ◦ σ ) /m !) Z$_{m}$ ( t ) as a consequence of the functional non-central limit theorem discussed in Section 2.1. The second term is o$_{P}$ (1) uniformly in t by Donsker’s theorem, if Var(log ε 2 $_{1}$) < ∞ . As a result, this observation is consistent with Corollary 3.2. 

Wilcoxon: Due to monotonicity of the logarithm, choosing ψ ( x ) = log( x $^{2}$) yields the same test statistic as if testing for a change in variance, i.e. if we choose ψ ( x ) = x $^{2}$. Moreover, the Wilcoxon test statistic is robust, i.e. it is designed to reduce the effect of heavy tails, so that it does not seem advisable to apply Wilcoxon-like tests when testing for a change in the tail index. 

## 5 Self-normalization 

An application of the CUSUM test presupposes knowledge of the normalizing sequence d$_{n,m}$ (if E( ψ ( X$_{1}$ ) | F $_{0}$) ̸ = 0) and of the coefficients J$_{m}$ (Ψ) or σ that appear in the limit of 

