Mathematica: Problem Computing Fisher Information of pdf

jcates7

Hello,

I'm trying to calculate Fisher information (and eventually the Cramer-Rao lower bound) for this particular pdf with Mathematica:

[itex]\text{pte}[t,\Theta ] = \frac{P_{\text{ec}}}{\tau _d-\tau _r}\left[e^{\frac{-(t-\Theta )}{\tau _d}}-e^{\frac{-(t-\Theta )}{\tau _r}}\right]; \text{domain}[\text{pte}] = \{t,-\infty,\infty\}\&\&\{\Theta >0\}[/itex]

So I want to find [itex]I(t|\Theta )[/itex]

This seems relatively straightforward with:

[itex]\text{Integrate}\left[D[\text{Log}[\text{pte}],\Theta ]^2,\{t,-\infty ,\infty \}\right][/itex]

However, Mathematica doesn't want to compute the integral. It just returns the integral itself:

[itex]\int_{-\infty}^{\infty} \frac{\left(\frac{e^{-\frac{t-\Theta }{\tau _d}}}{\tau _d}-\frac{e^{-\frac{t-\Theta }{\tau _r}}}{\tau _r}\right){}^2 \left(\frac{P_{\text{ec}}}{\tau _d-\tau _r}\right)'\left[e^{-\frac{t-\Theta }{\tau _d}}-e^{-\frac{t-\Theta }{\tau _r}}\right]{}^2}{\frac{P_{\text{ec}}}{\tau _d-\tau _r}\left[e^{-\frac{t-\Theta }{\tau _d}}-e^{-\frac{t-\Theta }{\tau _r}}\right]{}^2} \, dt[/itex]

Initially my thought is that there isn't a closed-form solution, but this is something I have seen calculated in journal papers with the same pdf. I'm not an experience Mathematica user. Is there something I have missed in the input (syntax or additional options) or are there any general simplifications that Mathematica would need?

Thanks!

Bill Simpson

I get a different integrand than you do:

In[1]:= pte[t_,omega_]:=Pec/(taud-taur)(E^-((t-omega)/taud)-E^-((t-omega)/taur));
i=D[Log[pte[t,omega]],omega]^2

Out[2]= (1/(E^((-omega + t)/taud)*taud) - 1/(E^((-omega + t)/taur)*taur))^2/(E^(-((-omega + t)/taud)) - E^(-((-omega + t)/taur)))^2

And if I do an indefinite integral I get

In[3]:= Integrate[i,t]

Out[3]= (-(E^(omega/taud + t/taur)*t*(taud^2 + taud*taur + taur^2)) + E^(t/taud + omega/taur)*(taud*(taud - taur)*taur + t*(taud^2 + taud*taur + taur^2)) - (E^(t/taud + omega/taur) - E^(omega/taud + t/taur))*taud*taur*(taud + taur)*Log[E^(t/taud + omega/taur) - E^(omega/taud + t/taur)])/((E^(t/taud + omega/taur) - E^(omega/taud + t/taur))*taud^2*taur^2)

Have I made a mess out of this somehow?

jcates7

No, that is the correct equation, and I get the same integrand when I copy it into my notebook. Is there an inherent mathematica function for me to now evaluate the integrand from the indefinite integral?

Should one expect, in general, for mathematica to have difficulties with definite integrals?

Thanks

Bill Simpson

Definite and indefinite integrals have many different special cases and complexities when dealing with complicated functions. IF we could assume non-negative and perhaps continuous then then we could just subtract the result of substituting your limits of integration. But Mathematica doesn't seem to be able to see what the limit will be without knowing the values for the rest of your coefficients.

What happens if you give plausible values to omega, taur and taud? Do you get sensible results for the definite, or indefinite, integrals then?

Sometimes Mathematica just needs a hint to be able to see how to find a solution. Do you know anything about the domains or relationships between your coefficients?