 Solution

The parameter-dependent part of the complete multinomial distribution log-likelihood isThe 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 isThe 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.