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Math Help - Parameter estimation

  1. #1
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    Parameter estimation

    Hi all

    I have two equations

     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

     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

     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  \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
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by chogo View Post
    Hi all

    I have two equations

     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

     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

     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  \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 \gamma which solves:

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


    RonL
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  3. #3
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    trapezium rule? like the code from numerical recipies in C
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by chogo View Post
    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
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  5. #5
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    just wanted to say thanks, the bisection method worked well.
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