# find the fisher information

#### amberxinya

$$\displaystyle X \ \ \ has \ \ \ distribution \ \ \ Exp(\lambda) \ \ \ \ p(x| \lambda)=\lambda exp(-\lambda x) \ \ \ \ \ \ and \ \ \ \ \phi=Prob(X>x|\lambda)$$

compute the fisher information for phi.

I think $$\displaystyle \phi=e^{-\lambda x}$$

and therefore the $$\displaystyle p(x|\phi)=\phi\frac{\log{\phi}}{-x}, \ \ \ L(x,\phi)=\log{\phi}+\log{\frac{ \log{\phi}}{-x}}$$

then the calculation becomes too complex.

#### Anonymous1

$$\displaystyle X \ \ \ has \ \ \ distribution \ \ \ Exp(\lambda) \ \ \ \ p(x| \lambda)=\lambda exp(-\lambda x) \ \ \ \ \ \ and \ \ \ \ \phi=Prob(X>x|\lambda)$$

compute the fisher information for phi.

I think $$\displaystyle \phi=e^{-\lambda x}$$

and therefore the $$\displaystyle p(x|\phi)=\phi\frac{\log{\phi}}{-x}, \ \ \ L(x,\phi)=\log{\phi}+\log{\frac{ \log{\phi}}{-x}}$$

then the calculation becomes too complex.
You want the second derivative w.r.t. phi?

#### Anonymous1

$$\displaystyle L(x,\phi)=\log{\phi}+\log{\log{\phi}}\underbrace{ - \log{(-x)}}$$ $$\displaystyle \color{red}{\text{[]EDIT[] You have to incorporate the -log(-x) part since:}}$$ $$\displaystyle \color{red}{\phi=e^{-\lambda x} \implies x = \frac{\log\phi}{-\lambda}}$$

$$\displaystyle \frac{d}{d\phi}L(x,\phi)=\frac{1}{\phi}+\frac{1}{\phi}\cdot\frac{1}{\log{\phi}}\color{red}{+...}$$

$$\displaystyle \frac{d^2}{d\phi^2}L(x,\phi)=-\frac{1}{\phi^2}+\left\{\frac{1}{\phi}\cdot\frac{d}{d\phi}[\frac{1}{\log{\phi}}] + \frac{1}{\log{\phi}}\cdot\frac{d}{d\phi}[\frac{1}{\phi}] \right\}\color{red}{+...}$$

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