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According to genetic theory, blood types MM, NM and NN should occur in a very large population with relative frequencies $\theta^{2},\ 2\theta(1-\theta),$ and $(1-\theta)^{2}$, where $\theta$ is the (unknown) gene frequency.

(a) Suppose that, in a random sample of size $n$ from the population, there are $x_{1},\ x_{2}$ and $x_{3}$ individuals of the three types respectively. Find an expression for $\hat{\theta}$ using moment method.

I know how to solve it by using ML, but I have no idea how to find a moment method estimator.

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Jakoer
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What makes this question tricky is that there is no explicit random variable specified. You have outcomes but no random variable attached to those outcomes; therefore, the notions of a sample mean, or an expectation, do not apply.

However, we can construct such a random variable: we note that if $\theta$ is the parameter of interest, then the probabilities of each outcome appear to correspond nicely to the number of "M" alleles observed. That is to say, we would let $X$ count the number of "M"s occurring in each outcome of blood type, and it follows from the above that $$X \sim \operatorname{Binomial}(2,\theta).$$ Then we can readily compute a method of moments estimator by noting that the sample mean is $$\bar x = \frac{2(x_1) + 1(x_2) + 0(x_3)}{n},$$ and we would equate this to the expectation of $X$: $$\operatorname{E}[X] = 2(\theta^2) + 1(2\theta(1-\theta)) + 0(1-\theta)^2 = 2\theta.$$ Then the method of moments estimator is the value of $\theta$ satisfying $$2\theta = \frac{2x_1 + x_2}{n}.$$

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