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Maximum likelihood estimate

  1. Feb 13, 2008 #1
    in need of help for how to do this question
    given probability mass function:
    x 1 2 3 4
    p(x) 1/4(θ+2) 1/4(θ) 1/4(1-θ) 1/4(1-θ)

    Marbles
    1=green
    2=blue
    3=red
    4=white

    For 3839 randomly picked marbles
    green=1997
    blue=32
    red=906
    white=904

    what is the max likelihood of θ using this data?
     
  2. jcsd
  3. Feb 13, 2008 #2

    EnumaElish

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    What is the likelihood function in this case?
     
  4. Feb 13, 2008 #3
    oops i left out that x=1,2,3,4 are of binomial distributions...
    would the likelihood function be the pmf of binomial dist.?
    = (nCx) p^x (1-p)^(n-x)

    and the loglikelihood function be:
    L(p)= log(nCx) + xlog(p) + (n-x)log(1-p) ??
     
  5. Feb 13, 2008 #4

    EnumaElish

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    Is it a binomial, or a multinomial distribution? Binomial has two possible outcomes; here you have four.
     
  6. Feb 13, 2008 #5
    i'm a little lost at this point, in the above section it says that for example green marbles is modelled by a r.v. N1 with a binomial (n, 1/4(θ+2)) distribution and blue is modelled by r.v. N2 with a binomial (n,1/4(θ)) dist. where n in both cases is total # of marbles (3839 in this case)

    so i'm assuming red and white have similar binomial dist.
     
  7. Feb 13, 2008 #6

    EnumaElish

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    It is possible to look at multinomial r.v.'s as a vector of binomial r.v.'s.

    The likelihood function (nCx) p^x (1-p)^(n-x) represents just one of the 4 variables, though (e.g., green vs. not green). To capture all individual colors you need to think in terms of a multinomial distribution with multiple (> 2) outcomes.
     
  8. Feb 13, 2008 #7
    hmm..so in this case i should use the multinomial prob. mass function to get the likelihood function.. then take the natural log of it correct?
    Do I differentiate now and how do I arrive at the estimate for theta?
     
  9. Feb 13, 2008 #8

    EnumaElish

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    You should set up the log likelihood function L, then differentiate it with respect to theta, set it to zero, and solve for theta: L'(θ) = 0 so θ* = L'-1(0). Then check L"(θ*) < 0 to make sure it's a maximum and not a minimum.
     
  10. Feb 13, 2008 #9
    thanks for the clarifications =)
     
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