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Thread: A problem of maximum likelihood estimator

  1. #1 A problem of maximum likelihood estimator 
    Forum Freshman
    Join Date
    Jul 2011
    Hi, everyone. I have a problem when reading the Population Genetics. The problem is as below:
    P and q are the possibility of two alleles. The possibility to get a sample that contains G individuals, with a AAs, b Aas, and c aas is
    [G!/a!b!c! ](p2) a (2pq) b (q2) c = [G! 2 b /a!b!c!] p 2a+b q b+2c
    to estimate the parameters of the sample, we have the log likelihood function:
    L = (2a + b) ln p + (b + 2c) ln q + constant
    dL/dp = (2a + b) / p (b+2c) / q = 0 ;
    ∴ p = (2a + b) / 2G
    1/ V (p) = -d2 L / dp2 = (2a + b) / p2 + (b + 2c) / q2 = 2G/ pq. (1)

    What confused me is that why the second derivative of the log likelihood function equals to the reciprocal of the variance of p?

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  3. #2  
    Forum Professor wallaby's Avatar
    Join Date
    Jul 2005
    I'm not very good at this sort of stuff, but i can give you some terms that an internet search might help elaborate on.

    the equality only occurs when the estimator (maximum likelihood estimator in this case) is "efficient" (term 1), for cases where the estimator is not efficient refer to the "Cramer-Rao Inequality" (term 2). Both of these cases assume that there is no bias in the estimation of the mle.

    I'll dig out my notes from last semester to be more specific if this hasn't helped or if no one else can explain.

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