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BOOTSTRAP AND JACKKNIFE METHODS IN EXTREMAL INDEX ESTIMATION 59
they propose a consistent estimation of Var⇣ b⇤n(b)⌘ and Bias⇣ b⇤n(b)⌘ given by d[b⇤ d⇣b⇤b⇤⌘
Varn ⌘ VARJAB( n(b)), Biasn = 2 n(b) n(2b) , [
where VARJAB is defined in (3.3). Parameters C1 and C2 can be consistently estimated by
b ⇣b⇤ b⇤ ⌘ C1 = nb VARJAB( n(b)), C2 = 2b n(b) n(2b) .
b 1[ b⇤
The NPPI estimator of the optimal block size is then given by
b0n = 2Cb2 1/3n1/3. ⇣ b2⌘
C1
The JAB methodology, allowing to assess the accuracy of bootstrap estima- tors for dependent data, was derived by Lahiri (2002). The key step is to delete resampled blocks instead of blocks of original data values.
If b⇤n(b) is the MBB estimator of n and ` = n b+1 the number of “observable” blocks of length b:
• Let m be an integer such that m ! 1 and m/n ! 0 as n ! 1, denoting the number of bootstrap blocks to be deleted.
• Write M = ` m+1 and for i = 1,···,M let us define the set Ii = {1,··· ,`}\{i,··· ,i+m 1}, denoting the index set of all blocks obtained by deleting the m blocks.
• Resample [n/b] from the reduced collection {Bj : j 2 Ii} and compute the ith jackknife block-deleted estimate, b⇤i ⌘ b⇤i(b), i = 1,··· ,M.
nn The JAB variance estimator for n(b) is then defined as
b⇤ X
mM
[ b⇤ e⇤i b⇤ 2
VARJAB( n(b)) = (` m)M ( n (b) n(b)) , (3.3) i=1
e⇤i b 1 h b⇤ b⇤i i
where n (b) = m ` n(b) (` m) n (b) is the ith block - deleted jackknife
pseudo-value of n(b), i = 1,··· ,M.
Figure 6 shows sample paths for ✓bUC estimates and the corresponding block
bootstrap estimates, for an initial block size and the “‘optimal” block size. According to suggestions given in Lahiri et al. (2007) to obtain Cb1 and Cb2, as a first approach we used b = c1n1/5, m = c2n1/3b2/3 with c1 = 1 and c2 = 1.