Can I use another parameter in an unbiased estimator?

Question

Given a multinomial distribution, the task is to make an unbiased estimator for one parameter. However, the parameters all add up to 1, so how can I make an unbiased estimator without using other parameters. Am I allowed to use other parameters?

Answer 1

Think about a binomial for simplicity. Probabilities are $p$ and $q$, respectively. Then $p+q=1$. In principle if you know $q$ then you know $p$, and using that the expectation is linear then given an unbiased estimator for $q$ you would have an unbiased estimator for $p$ as well. In that sense, $p$ and $q$ are not independent. The same happens with your multinomial distribution. In that case an unbiased estimator for $L_1$ would be the sample proportion of observations where $X=1$. That is

$\hat{L}_1=\sum 1_{X(i)=0}/n$

Notice that $1_{X(i)=0}$ has a Bernoulli distribution with probability $L_1$ and the rest follows from there.