MLE estimation in genetic experiment

Question

Consider the following problem:

In a biology experiment, we list the observed frequencies of 4 genetic classes and assume the genetic probability of each class. The information is given in the table below:

\begin{array}{|c|c|c|c|} \hline Genetic \; class & Observed \; frequency & Probability \\ \hline AB & n_1 & \frac{2+\theta}{4} \\ \hline Ab & n_2 & \frac{1-\theta}{4} \\ \hline aB & n_3 & \frac{1-\theta}{4} \\ \hline ab & n_4 & \frac{\theta}{4} \\ \hline \end{array}

I preliminary observation is that $\theta \in [0,1]$.

1. Determine the MLE of $\theta$.

2. For the case of $n_i \to \infty$ compute the variance of the MLE.

My problems

I have problems with the first part since there is not a single functional form. The equation depends on the value. Should I infer a proper distribution?

I'd say the second part follows for some asymptotic result but without the first I cannot proceed.

References

Apparently, this is a variation of problem 3.18 in Examples in Parametric Inference with R by Ulhas Jayram Dixit.

My progress

1. Assuming a multinomial distribution I get that $f(\theta|x) \propto \Big(\frac{2+\theta}{4}\Big)^{n_1}\Big(\frac{1-\theta}{4}\Big)^{n_2+n_3}\Big(\frac{\theta}{4}\Big)^{n_4}$ with $\theta \in ]0,1[$.

Then the likelihood equation gives $\frac{d}{d\theta} log f(\theta|x) = \frac{n_1}{2+\theta}-\frac{n_2+n_3}{1-\theta}+\frac{n_4}{\theta}$. Setting this to zero gives (as valid value in ]0,1[) $\theta = -1 + \sqrt{(\frac{n_1-2n_2-2n_3-n_4}{4n_4})^2 + \frac{n}{2n_4}}$. The second derivative gives an always negative value so this must be a relative maximum. Given that in the extremes the likelihood is zero, the value has to be a global maximum.

2. Asymptotic normality of mle tells me that $\hat \theta_n \approx N(\theta_0,\frac{1}{I_{\overline{x}}(\theta_0)})$ where $\theta_n$ is the mle sequence and $\theta_0$ is the real value of the parameter. I stress that its variance is the information of the whole sample vector. Since $I_{\overline{x}}(\theta_0) = Var_{X|\theta_0} S(X|\theta_0)$, I wonder if it is legal to approximate $Var_{X|\theta_0} S(X|\theta_0) \approx Var_{X|\hat \theta_n} S(X|\hat \theta_n)$ as the value of $\theta_0$ is unknown...

Answer 1

If I told you that $X$ follows a categorical distribution with $m = 4$ categories with probabilities $(p_1, p_2, p_3, p_4)$--that is to say, $$\Pr[X = i] = \begin{cases} p_i, & i \in \{1, 2, 3, 4\}, \\ 0, & \text{otherwise}, \end{cases}$$ then how would you find the MLE of, say, $p_1$ for a sample $\boldsymbol X$?

The same idea applies to your case, except instead of estimating a single parameter out of (essentially) three parameters (the fourth is determined from the others), all four categorical probabilities are related through the single parameter $\theta$. Consequently, the first thing to do is to write the likelihood function for $\theta$ given the sample.

Note that for any categorical distribution, a sufficient statistic for the joint probability parameter vector $\boldsymbol p = (p_1, \ldots, p_m)$ is the frequency of the respective categories observed in your sample. For example, if there are four categories, then the sample $$\boldsymbol x = (1, 2, 1, 1, 3, 1, 2, 2, 3, 1, 4)$$ can be reduced to $$\boldsymbol T(\boldsymbol x) = (5, 3, 2, 1),$$ where the $i^{\rm th}$ position counts the number of occurrences of the $i^{\rm th}$ categorical outcome. You may be able to achieve even further data reduction when the categorical probabilities are related through $\theta$, but $\boldsymbol T$ is a start for you to figure out how to write $$\mathcal L(\theta \mid \boldsymbol x) = \mathcal L (\theta \mid \boldsymbol T(\boldsymbol x)).$$