1. ## Parameter estimation

Hi all

I have two equations

$\displaystyle F(m) = \int_0^1 \frac{1-x^m - (1-x)^m}{1-x} \cdot \frac{1-e^{-2\gamma x}}{2\gamma x} dx$

and

$\displaystyle G(m) = \int_0^1 x^{m-1} \cdot \frac{1-e^{-2 \gamma (1-x)}}{2\gamma (1-x)}$

i then have a relation

$\displaystyle Z = \frac{F(m) + F(n)}{G(m) + G(n)}$

Where m, and n are integers which i know.
Z is a constant which i know.
and i want to find a value of $\displaystyle \gamma$ which will satisfy the above equations

Any ideas? Should i use a numerical integration procedure to evaluate the integrals then solve? Should i do it manually? (if so on inspection will this take a year to solve) or should i use some sort of Likelihood method of parameter evaluation

Thanks for the time you spend helping. Its an invaluable resource

chogo

2. Originally Posted by chogo
Hi all

I have two equations

$\displaystyle F(m) = \int_0^1 \frac{1-x^m - (1-x)^m}{1-x} \cdot \frac{1-e^{-2\gamma x}}{2\gamma x} dx$

and

$\displaystyle G(m) = \int_0^1 x^{m-1} \cdot \frac{1-e^{-2 \gamma (1-x)}}{2\gamma (1-x)}$

i then have a relation

$\displaystyle Z = \frac{F(m) + F(n)}{G(m) + G(n)}$

Where m, and n are integers which i know.
Z is a constant which i know.
and i want to find a value of $\displaystyle \gamma$ which will satisfy the above equations

Any ideas? Should i use a numerical integration procedure to evaluate the integrals then solve? Should i do it manually? (if so on inspection will this take a year to solve) or should i use some sort of Likelihood method of parameter evaluation

Thanks for the time you spend helping. Its an invaluable resource

chogo
Use a numerical method to find $\displaystyle \gamma$ which solves:

$\displaystyle Z - \frac{F_{\gamma} (m) + F_{\gamma}(n)}{G_{\gamma}(m) + G_{\gamma}(n)}=0$

RonL

3. trapezium rule? like the code from numerical recipies in C

4. Originally Posted by chogo
trapezium rule? like the code from numerical recipies in C
Bisection method. Use numerical integration to evaluate the integrals with whichever method you are most at home with.

RonL

5. just wanted to say thanks, the bisection method worked well.