MLE of multinomial distribution There are three possible outcomes of an experiment: $A$, $B$, and $C$ but in some cases we cannot distinguish $A$ from $B$. Let $p_a , p_b$ and $p_c$ be respectively probabilities of $A$, $B$, and $C$.
After $n$ independent experiments $A$ happened $n_a$ times, $B - n_b$ times and $C - n_c$ times but $n_a+n_b+n_c <n$ (in other cases we could not tell if the result is $A$ or $B$).
Find MLE of $p_a, p_b, p_c$.
Here are some of my ideas:


*

*$(A,B,C) \sim \operatorname{Mult}(n, p_a, p_b, p_c)$ but in this situation $n_a+n_b+n_c<n$ so I am not sure if it is justified to use density function in its classic form.

*Computing $p_c$ would not be a problem if we consider a model:
$(X,C) \sim \operatorname{Mult}(n, 1-p_c , p_c)$ where $X= A \cup B$ that occured $n-n_c$ times.
I would appreciate any hint. Especially for computing $p_a$ and $p_b$. 
 A: You have $n_a$ observations in which the outcome is known to be $A$, which has probability $p_a$.
You have $n_b$ observations in which the outcome is known to be $B$, which has probability $p_b$.
You have $n_c$ observations in which the outcome is known to be $C$, which has probability $1-p_a-p_b$.
You have $n-n_a-n_b-n_c$ observations in which the outcome is known to be  either $A$ or $B$, which has probability $p_a+p_b$.
The likelihood is therefore
$$
L(p_a,p_b) = \text{constant}\times p_a^{n_a} p_b^{n_b} (1-p_a-p_b)^{n_c} (p_a+p_b)^{n-n_a-n_b-n_c},
$$
where "constant" means not depending on $p_a,p_b$. So
\begin{align}
& \ell(p_a,p_b) = \log L(p_a,p_b) \\[10pt]
= {} & \text{constant} + n_a\log p_a + n_b \log p_b \\
& {} + n_c \log (1-p_a-p_b) + (n-n_a-n_b-n_c) \log (p_a+p_b),
\end{align}
and
$$
\frac \partial {\partial p_a} \ell(p_a,p_b) = \frac{n_a}{p_a} - \frac{n_c}{1-p_a-p_b} + \frac{n-n_a-n_b-n_c}{p_a+p_b} 
$$
and the derivative with respoect to $p_b$ is found similarly.
Both partial derivatives should vanish at the maximum point.
Since the domain of $L$ is a compact set (given by $p_a\ge0$, $p_b\ge 0$, $p_a+p_b\le 1$) and $L$ is zero on the boundary and positive in the interior, there has to be at least one global maximum point. If there's only one point where the derivatives vanish, then that's it.
