next up previous contents
Next: Estimation of the scaling Up: The identification of strange Previous: Information from outside the   Contents

The variance of the entropy estimator

To obtain the variances of the dimension and entropy estimators, without reparametrization, we have to invert the information matrix
\begin{displaymath}
I = \left(
\begin{array}{lll}
I_{\nu\nu} & I_{\nu\phi_{d}} &...
...d+e}\phi_{d}} & I_{\phi_{d+e}\phi_{d+e}}\\
\end{array}\right)
\end{displaymath} (C.1)

where
$\displaystyle I_{\nu\nu}$ $\textstyle =$ $\displaystyle -\mbox{E} \left\{
\frac{\partial^{2} \ln L(\nu , \phi_{d} , \phi_...
...u^{2}} +
\frac{N_{d+e}\phi_{d+e}(r_{u,d+e}^{\nu} -
r_{l,d+e}^{\nu})}{\nu^{2}} +$  
    $\displaystyle N_{d}\phi_{d}r_{u,d}^{\nu} \ln^{2}(r_{u,d}) +
\frac{N_{d}\phi_{d}^{2}r_{u,d}^{2\nu}\ln^{2}(r_{u,d})}
{1 - \phi_{d}r_{u,d}^{\nu}} +$  
    $\displaystyle N_{d+e}\phi_{d+e}r_{u,d+e}^{\nu}\ln^{2}(r_{u,d+e}) +
\frac{N_{d+e}\phi_{d+e}^{2}r_{u,d+e}^{2\nu}\ln^{2}(r_1,d+e)}
{1 - \phi_{d+e}r_{u,d+e}^{\nu}}$ (C.2)


\begin{displaymath}
I_{\nu\phi_{d}} = I_{\phi_{d}\nu} = -\mbox{E} \left\{
\frac{...
...hi_{d}r_{u,d}^{2\nu}\ln(r_{u,d})}
{1 - \phi_{d}r_{u,d}^{\nu}}
\end{displaymath} (C.3)


\begin{displaymath}
I_{\phi_{d}\phi_{d}} = -\mbox{E} \left\{
\frac{\partial^{2}\...
...\frac{N_{d}r_{u,d}^{\nu}}{\phi_{d}(1 - \phi_{d}r_{u,d}^{\nu})}
\end{displaymath} (C.4)


\begin{displaymath}
I_{\phi_{d}\phi_{d+e}} = 0
\end{displaymath} (C.5)

Note that
\begin{displaymath}
I_{\phi_{d}\phi_{d}} = \frac{I_{\nu\phi_{d}}}{\phi_{d}\ln(r_{u,d})}
\end{displaymath} (C.6)

The equations are similar for the derivatives for $\phi_{d+e}$. Write
\begin{displaymath}
I_{\nu\nu} = A_{\nu} + \phi_{d}\ln(r_{u,d})I_{\nu\phi_{d}} +...
...d}}} +
\frac{I_{\nu\phi_{d+e}}^{2}} {I_{\phi_{d+e}\phi_{d+e}}}
\end{displaymath} (C.7)

where we used the abbreviation
\begin{displaymath}
A_{\nu} = \frac{N_{d}\phi_{d}(r_{u,d}^{\nu} - r_{l,d}^{\nu})...
...{d+e}\phi_{d+e}(r_{u,d+e}^{\nu} - r_{l,d+e}^{\nu})}
{\nu^{2}}
\end{displaymath} (C.8)

The determinant of the confusion matrix is:
$\displaystyle \mbox{det}(I)$ $\textstyle =$ $\displaystyle I_{\nu\nu}I_{\phi_{d}\phi_{d}}I_{\phi_{d+e}\phi_{d+e}} -
I_{\phi_...
...}\nu}^{2}I_{\phi_{d+e}\phi_{d+e}} -
I_{\nu\phi_{d+e}}^{2}I_{\phi_{d}\phi_{d}} =$  
    $\displaystyle \left( A_{\nu} + \frac{I_{\nu\phi_{d}}^{2}}{I_{\phi_{d}\phi_{d}}}...
..._{d}}^{2}I_{\phi_{d+e}\phi_{d+e}} -
I_{\nu\phi_{d+e}}^{2}I_{\phi_{d}\phi_{d}} =$  
    $\displaystyle A_{\nu}I_{\phi_{d}\phi_{d}}I_{\phi_{d+e}\phi_{d+e}}$ (C.9)

Now
\begin{displaymath}
\mbox{VAR}\left\{\hat{\nu}\right\} = I_{\nu\nu}^{-1} =
\frac...
...) +
N_{d+e}\phi_{d+e}(r_{u,d+e}^{\nu} - r_{l,d+e}^{\nu})} \\
\end{displaymath} (C.10)

and
$\displaystyle \mbox{VAR}\left\{\hat{\phi}_{d}\right\}$ $\textstyle =$ $\displaystyle I_{\phi_{d}\phi_{d}}^{-1} =
\frac{I_{\nu\nu}I_{\phi_{d+e}\phi_{d+...
... I_{\nu\phi_{d+e}}^{2}}
{A_{\nu}I_{\phi_{d}\phi_{d}}I_{\phi_{d+e}\phi_{d+e}}} =$  
    $\displaystyle \frac{1}{I_{\phi_{d}\phi_{d}}} +
\frac{\phi_{d}^{2}\ln^{2}(r_{u,d+e})}{A_{\nu}}$ (C.11)

And a similar expression for $\mbox{VAR}\left\{\hat{\phi}_{d+e}\right\}$. Note that these latter variances depend on the reference distance. The estimator of the entropy becomes:
\begin{displaymath}
\hat{K}_{2} = \frac{\ln\left(\frac{\hat{\phi}_{d}}{\hat{\phi...
...{N_{d+e}r_{u,d+e}^{\nu}}{N_{l,d+e}+N_{s,d+e}}\right)}
{e\tau}
\end{displaymath} (C.12)

With
\begin{displaymath}
T = \frac{\ln\left(\frac{\phi_{d}}{\phi_{d+e}}\right)}{e\tau}
\end{displaymath} (C.13)

the variance of the entropy estimator becomes
\begin{displaymath}
\mbox{VAR}\left\{\hat{K}_{2}\right\} =
\frac{
\frac{I_{\phi_...
...{\phi_{d}\phi_{d+e}}^{-1}}{\phi_{d}\phi_{d+e}}}
{(e\tau)^{2}}
\end{displaymath} (C.14)

Using the expressions for $\hat{\phi}_{d}$, $\hat{\phi}_{d+e}$ and
\begin{displaymath}
I_{\phi_{d}\phi_{d+e}}^{-1} =
\frac{I_{\nu\phi_{d}}I_{\nu\ph...
...
I_{\phi_{d}\phi_{d}}I_{\phi_{d+e}\phi_{d+e}}}{\mbox{det}(I)}
\end{displaymath} (C.15)


$\displaystyle \mbox{VAR}\left\{\hat{K}_{2}\right\}$ $\textstyle =$ $\displaystyle \frac{\frac{1}{I_{\phi_{d}\phi_{d}}} +
\frac{\phi_{d}^{2}\ln^{2}(...
...+e}^{2}(e\tau)^{2}} -
2.\frac{\ln(r_{u,d})\ln(r_{u,d+e})}{A_{\nu}(e\tau)^{2}} =$  
    $\displaystyle \frac{\frac{1-\phi_{d}r_{u,d}^{\nu}}{N_{d}\phi_{d}r_{u,d}^{\nu}} ...
... +
\frac{\ln^{2}\left(\frac{r_{u,d}}{r_{u,d+e}}\right)}{A_{\nu}}}
{(e\tau)^{2}}$ (C.16)


next up previous contents
Next: Estimation of the scaling Up: The identification of strange Previous: Information from outside the   Contents
webmaster@rullf2.xs4all.nl