Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 40 / 1 / b / v / Solution

Differentiate the log-likelihood again. For $d>0$, the positive interior maximum-likelihood estimate givesThe observed information is $-{\ell }^{\prime \prime }(\hat{\theta })=d/{\hat{\theta }}^2$, giving the usual local standard error approximation $\hat{\theta }/\sqrt{d}$. For $d=0$ there is no positive interior maximum, so the displayed substitution and this standard error approximation do not apply; at every positive $\theta$ the second derivative is zero.