is the sign of the number of element in the set. In practice, we apply the series of tests *T* (1) , *T* (2) , *T* (3) , ......... , in this order until the null hypothesis is not rejected. We used the previous methods, to derive the distribution of the inter-event distances. The results obtained for the aftershock sequence triggered by the Al Hoceima 1994 earthquake are shown on Fig. 11.

the alternative *H*<sup>1</sup>

The test *T* (*<sup>k</sup>* )

hypothesis *H*<sup>0</sup>

(*k*).

*h* < *hcrit*

The test *T* (*<sup>k</sup>* )

Efron formula,

*x* \_

if *hcrit*

*T* (1) , *T* (2) , *T* (3)

*k*

**Theorem** (Silverman, 1981)

The kernel density estimate *f*

<sup>=</sup> (*x*<sup>1</sup> , *<sup>x</sup>*<sup>2</sup> ,........., *xn*); *x*¯ and *<sup>σ</sup>* <sup>2</sup>

\* (*k*) is the parameter obtained from *x*

*k*

parameter of the Gaussian kernel density estimate *f*

66 Earthquake Research and Analysis - New Advances in Seismology

: " *f* has more than *k* bumps". Under the hypothesis *H*<sup>0</sup>

is inspired from the fact that big values of parameter *hcrit*

is constructed by simulating *N* statistics *hcrit*

**Proof.** The proof of this theorem is given in details in Silverman (1981).

*h y x x xh k*


s

is a randomly simulated sample from the standard normal distribution.

*value*

*P*

from *N* smoothed bootstrap samples of size *n* obtained from *x*<sup>1</sup> , *x*<sup>2</sup> ,........., *xn*.

. The following therorem by Silverman (1981) gives a characterisation

Under the null hypothesis, samples can be simulated from the kernel density estimate by using

where (*xI* (*i*); *i* =1 , n) is a bootstrap sample of size *n* simulated from the sample

{ ( ) ( ) } ( ) # *<sup>i</sup> crit crit*

*N*
