= Solution
The parameter-dependent part of the complete <multinomial distribution> <log-likelihood> is
$$
\ell_c(\theta)=y_1\log(1-\theta)+(y_2+y_3)\log\theta+C(y).
$$
The remaining probability factors and multinomial coefficient are constant in $\theta$. If $y_1+y_2+y_3>0$, differentiating gives $-y_1/(1-\theta)+(y_2+y_3)/\theta$, whose zero is
$$
\boxed{\widehat\theta=\frac{y_2+y_3}{y_1+y_2+y_3}.}
$$
The <log-likelihood> is concave. If $y_2+y_3=0<y_1$, its maximum is at zero; if $y_1=0<y_2+y_3$, its maximum is at one. These boundary cases are included by the same formula. If $y_1=y_2=y_3=0$, the data lie entirely in the parameter-independent fourth cell and every $\theta\in[0,1]$ is a <maximum-likelihood estimator>.
Back to article page
