= Solution
Set $S=n_{11}+n_{22}$, $R=n_{12}+n_{21}$ and $N=S+R$. With equal positive rates, the one-year <probability> of a change is $q(\lambda)=(1-e^{-2\lambda})/2\in(0,1/2)$, so the conditional <likelihood function> is proportional to
$$
(1-q)^Sq^R.
$$
For $N>0$, its unrestricted binomial maximizer is $q=R/N$. A finite interior positive-rate maximum exists exactly when $0<R/N<1/2$, or
$$
\boxed{0<R<S,\qquad \widehat\lambda=\frac12\log\frac{S+R}{S-R}.}
$$
Indeed the binomial <log-likelihood> is strictly concave at this interior maximum and $q(\lambda)$ is strictly increasing. If $R\geq S$ and $R>0$, the likelihood increases towards its supremum as $q\uparrow1/2$, requiring $\lambda\to\infty$; no finite maximum exists. Equality $R=S>0$ also has only this limiting maximum.
If $R=0<S$, allowing the boundary $\lambda=0$ gives a finite maximum there; insisting on $\lambda>0$ gives only a supremum as $\lambda\downarrow0$. Thus if “valid” includes zero rates, the condition is $R<S$ with $N>0$; if rates must be strictly positive, it is $0<R<S$. If $N=0$, every rate has the same empty-data likelihood and the <statistical parameter> is not identifiable. These boundary qualifications prevent a negative or infinite plug-in rate from being called a valid interior estimate.
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