Past exam of the options I course of the University of Oxford / 2012 / O / O1 / b

Let $z={\Phi }^{-1}(1-\alpha /2)$ for the standard normal cumulative distribution function. Consistency of the maximum-likelihood estimator and Slutsky's theorem give the asymptotic standardizationIts probability of lying between $-z$ and $z$ tends to $1-\alpha$. Solving those inequalities for the unknown parameter yields the Wald confidence intervalThus the estimated standard error is $I_n(\hat{\theta }{)}^{-1/2}$. Its coverage is approximate and relies on the large-sample regularity assumptions.