next up previous contents
Next: The number of classes Up: Goodness-of-fit tests Previous: Goodness-of-fit tests   Contents

The equal-probabilities method of constructing classes

The computed distances within the scaling region $]r_{l},r_{u}]$, should preferably be grouped in $k$ classes using the equal-probabilities method [Kendall and Stuart, 1979, recommendation p.465]. So we would like to have
\begin{displaymath}
\int_{r_{l}}^{r_{i}} \frac{\nu r^{\nu-1}} {r_{u}^{\nu}-r_{l}...
...c{i}{k}, \quad i \in [0,k] \;\mbox{with}\; r_0=r_l; \; r_k=r_u
\end{displaymath} (5.60)

where $r_{i}$ are the class boundaries. Hence, they must be chosen according to
\begin{displaymath}
r_{i} = { \left( \frac{i(r_{u}^{\nu}-r_{l}^{\nu})}{k} +
r_{l}^{\nu} \right) }^{1/\nu}
\end{displaymath} (5.61)

So we can construct optimal class boundaries if we know (or estimate) the dimension.



webmaster@rullf2.xs4all.nl