Method of moments/maximum likelihood

[Gémeaux]

I just needed some help with a few questions.

Consider N independent random variables having identical binomial distributions with parameters Θ and n= 3. If n0 of them take on the value 0, n1 of them take on the value 1, n2 of them take on the value 2 and n3 of them take on the value 3, use the method of moments to find a formula for estimating Θ.

Since X=nΘ, therefore X=3Θ. Now we need to find the mean of the random variables, which is (0*n0+1*n1+2*n2+3*n3)/N, since there are N random variables. So we get 3Θ = (n1+2n2+3n3)/N. Hence, Θ=(n1+2n2+3n3)/3N

Could someone tell me if I'm doing this right? Also, how would you use the method of maximum likelihood to estimate Θ?

Thanks in advance.

 

[Valkarie]

Yes, you are doing this correctly. For the method of maximum likelihood, we need to find the value of Θ that maximizes the likelihood function. The likelihood function for this problem is given by: L(θ) = n0(1-θ)^3 + n1(1-θ)^2θ + n2(1-θ)θ^2 + n3θ^3. To find the value of Θ that maximizes L(θ), we need to differentiate L(θ) and set it equal to 0. This will give us a quadratic equation whose roots will give us the value of Θ that maximizes the likelihood. The solution of this equation will result in an estimate of Θ.

 

[tacman]

Yes, you are on the right track with using the method of moments to estimate Θ. Another way to think about it is to equate the theoretical mean of the binomial distribution, which is np, to the sample mean, which is (n1+2n2+3n3)/N. So we get np = (n1+2n2+3n3)/N, which gives us the same formula as before for estimating Θ.

To use the method of maximum likelihood to estimate Θ, we need to first write out the likelihood function, which is the probability of observing the data given the parameter Θ. In this case, it would be the product of the binomial probabilities for each of the possible outcomes (0, 1, 2, and 3). So the likelihood function would be L(Θ) = (Θ^0)(1-Θ)^n0 * (Θ^1)(1-Θ)^n1 * (Θ^2)(1-Θ)^n2 * (Θ^3)(1-Θ)^n3.

To find the maximum likelihood estimate of Θ, we would take the derivative of the likelihood function with respect to Θ, set it equal to 0, and solve for Θ. This would give us the value of Θ that maximizes the likelihood of observing the given data. However, in this case, the derivative is not easy to solve analytically, so we can use numerical methods or software to find the maximum likelihood estimate of Θ.

Overall, both the method of moments and the method of maximum likelihood can be used to estimate parameters in a statistical model. The method of moments is simpler and easier to understand, but the method of maximum likelihood is more powerful and can handle more complex models.