"Approximation" of a maximum likelihood confidence set

Question

I have some trouble constructing the "approximate" set, how can it be defined or calculated? Suppose that three characteristics in a large population can be observed according to the following frequencies $p_1= µ^3,\ p_2= 3µ(1-µ),\ p_3= (1-µ)^3$ where $0<\mu<1$. Let $N_j; j = 1; 2; 3$ be the observed frequencies of characteristic $j$ in a random sample of size $n$. Construct the approximate level $(1-\alpha)$ maximum likelihood confidence set for $\mu$.

Answer 1

Your model is a one-parameter model ($\mu$), so you want to maximize the likelihood of observing your sample by adjusting $\mu$:

$L(\mu|N_j) = (\mu^{3N_1})\times [3\mu(1-\mu)]^{N_2}\times(1-\mu)^{3N_3}$

Take logarithms of both sides to make it easier to work with:

$ \ln(L(\mu|N_j))=\mathcal{L}(\mu|N_j) = 3N_1\ln(\mu)+N_2[\ln(3)+\ln(\mu)+\ln(1-\mu)]+3N_3\ln(1-\mu)$

The MLE will be the value of $\mu$ that makes the derivative of $\mathcal{L} = 0$:

$\frac{d}{d\mu} \mathcal{L}(\mu|N_j) = \frac{3N_1}{\mu}+\frac{N_2}{\mu}-\frac{N_2}{1-\mu}-\frac{3N_3}{1-\mu} = \frac{3N_1+N_2}{\mu}-\frac{N_2+3N_3}{1-\mu}=0$

Therefore, the MLE is $\mu: \frac{1-\mu}{\mu} = \frac{N_2+3N_3}{3N_1+N_2}\rightarrow \hat \mu=\left(\frac{N_2+3N_3}{3N_1+N_2}+1\right)^{-1}$.

Now, to answer your question on the approximate confidence set for $\mu$:

1. Calculate $L^*=L(\hat \mu|N_j)$
2. Form the likelihood ratio: $LR = \frac{L(\mu|N_j)}{L^*}$
3. Rely on asymptotic Chi-squared distribution of LR to form a CI. The $\mu$ that make the LR fall within the chi-squared bounds will be your CI.