Maximum likelihood estimate

1. Feb 13, 2008

icedsake

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. Feb 13, 2008

EnumaElish

What is the likelihood function in this case?

3. Feb 13, 2008

icedsake

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) ??

4. Feb 13, 2008

EnumaElish

Is it a binomial, or a multinomial distribution? Binomial has two possible outcomes; here you have four.

5. Feb 13, 2008

icedsake

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.

6. Feb 13, 2008

EnumaElish

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.

7. Feb 13, 2008

icedsake

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?

8. Feb 13, 2008

EnumaElish

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.

9. Feb 13, 2008

icedsake

thanks for the clarifications =)