Page 215 - Special Topic Session (STS) - Volume 2

STS486 Tonio D.B. et al.
where () is normally distributed with mean 0 and covariance matrix Σ ,

is the smoothing parameter and Σ is the covariance matrix of
∗
∗
(), … . , (). We indicate with ∗ = ( (), … . , ()) the bootstrap

1

1
sample statistic of the generic b-th replication, with b=1,…,B. The ordering of
the bootstrap statistic is obtained by introducing a suitable distance,
((), ()), using e.g.:
∗
2
2
∗
- -metric: (∫(() − ) ) 1/2
∗
∞
∗
- The supremum -metric : ||() − ()|| = |() − ()|

Thus, it is easy to define the (1 − )% bootstrap confidence band such that
((), ()) ≤ with the (1 − )% quantile of the distances between
∗

the bootstrap re-samples and the sample estimate. As shown by Cuevas et al.
(2006), the performances of the bootstrap approximations via extensive
simulations and real data applications provides asymptotic validity results
without the strong assumption of normality.

3. Result: Application to the Rivers of Lazio
As an example, the method illustrated in Section 2 is applied to a real
dataset concerning fish biodiversity of Lazio’s current waters. The data set is
available at the website http://dati.lazio.it/catalog/it/dataset/bioittica and
consists of fish abundances of 54 species detected in 33 rivers of Lazio (Fig. 1)
in 2015.

Figure 1: Map of Lazio’s watercourses (http://www.arpalazio.gov.it/)

To evaluate fish biodiversity in this area, a diversity profile approach is
adopted, choosing the Hill’s number (Hill, 1973):

= (∑ =1 ) 1/(1−) (6)

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