 Events divided by exposure estimator

For independent exponentially distributed failure times subject to independent right censoring, let $d$ be observed failures and $E={\sum }_i{x}_i$ total follow-up exposure. The parameter-dependent likelihood is ${\theta }^d{e}^{-\theta E}$, giving maximum-likelihood estimator $\hat{\theta }=d/E$. The fitted individual cumulative hazards sum to $d$, but this estimating identity alone does not establish that a constant hazard model fits.