Page 68 - Textos de Matemática Vol. 47

58 M. M. NEVES
bootstrap estimation of characteristics of an estimator. Two of these charac- teristics are the bias and the variance of the estimator. This method employs non-parametric resampling procedures to estimate the relevant constants in the leading term of the “optimal” block length, so it does not require the knowledge and/or derivation of explicit analytical expressions for those constants.
Extremal index estimators usually have a high bias. In most cases the bias is the main component of the MSE. As a first step it was thought to control the bias through a bootstrap estimator, that needs an adequate block size choice for resampling. The method applied here is based on the Jackknife-After-Bootstrap (JAB) of Efron (1992) and Lahiri (2002) to estimate the relevant constants that appear in expansions of the bias of the bootstrap estimator, in order to obtain a more stable and accurate estimate path.
3.2. The NPPI procedure. Given the observations (X1,X2,...,Xn), let ⇥bn be an estimator of ✓. Let us consider now the bias of ⇥bn
n ⌘Bias⇣⇥bn⌘=E⇣⇥bn⌘ ✓,
and the corresponding block bootstrap estimator based on blocks of size b (we
shall consider here the MBB),
b ⇤ ( b ) = E ⇣ ⇥b ⇤ ( b ) ⌘ ✓b ,
where E⇤ denotes the conditional mean, given the data.
Lahiri et al. (2007), Section 2, remember that for many population param-
eters (denoted here by n), the variance of the corresponding block bootstrap estimator is an increasing function of the block size, b, while its bias is a decreas- ing function of b, and that under suitable regularity conditions, the variance and the bias of the block bootstrap estimator admit expansions of the form
n2a Var⇣ b⇤n(b)⌘ = C1n 1b+o n 1b , (3.1)
na Bias ⇣ ⇤n(b)⌘ = C2b 1 + o b 1 , (3.2)
n⇤nn
as n ! 1 and over a suitable set n ⇢ {2,··· ,n} of possible block length b. C1, C2 and a are constants depending on the characteristics under study. Hall et al. (1995) and Lahiri et al. (2007) showed that the optimal block size b0n has the form
b0n = ⇣2C2 ⌘1/3n1/3(1 + o(1)). C1
Now, for obtaining
C1 ⇠nb 1Var⇣ b⇤n(b)⌘ and C2 ⇠bBias⇣ b⇤n(b)⌘,