Likelihood function for MLE

Question

Refer to the Example 7 in this lecture:

How did the author obtain the likelihood function ? Is it from binomial? Can someone show the steps to the likelihood function? Thank you!

Answer 1

Suppose $k_1+k_2+k_3=n$ with $k_i\in\{0,1,\ldots,n\}$, so that a sample of $n$ people is being considered. Here $k_i$ denotes the observed number of people in the sample having a particular genotype.

Let $X_i$ be the number of people having a particular genotype in the population, $i=1,2,3$.

Then $(X_1,X_2,X_3)$ has a multinomial distribution with probability mass function

$$P(X_1=k_1,X_2=k_2,X_3=k_3\mid\theta)=\frac{n!}{k_1!k_2!k_3!}\theta^{2k_1}(2\theta(1-\theta))^{k_2}(1-\theta)^{2k_3}\quad,k_1+k_2+k_3=n$$

This is the formula for the likelihood function used in your note.

But since the likelihood is a function of the parameter $\theta$ with respect to which it is to be maximized to get an estimate of $\theta$, you can drop the constant which is free of $\theta$.

So having observed $(k_1,k_2,k_3)$, likelihood function is simply

$$L(\theta\mid k_1,k_2,k_3)\propto \theta^{2k_1}(2\theta(1-\theta))^{k_2}(1-\theta)^{2k_3}\quad,\,0<\theta<1$$

Answer 2

This is a straightforward application of the hypergeometric distribution, the generalization of the binomial distribution to more than two categories.