Large-Sample Wald and Likelihood-Ratio Intervals

The previous lesson's exact intervals were a luxury: they existed because the normal distribution supplied clean, tabulated pivots — $t(n-1)$ and $\chi^2(n-1)$ — with no free parameters left in them. A Poisson jump-intensity, a Bernoulli default probability, a Gamma shape parameter, a GARCH persistence coefficient: none of these come with an off-the-shelf exact pivot, and deriving one by hand is often intractable. Asymptotic Properties of the MLE already proved the fix applies almost universally: any regular MLE satisfies $\sqrt n(\hat\theta_n-\theta_0)\to_d\mathcal N(0,1/I_1(\theta_0))$, which is itself an approximate pivot — not exact for any finite $n$, but converging to something parameter-free as $n$ grows. This note turns that single theorem into three competing large-sample interval constructions — the Wald interval (the direct, plug-in version), the likelihood-ratio/profile interval (built by inverting the same generalized-likelihood-ratio statistic that will reappear as a hypothesis test), and the delta-method interval for a transformed parameter — and is explicit about the one thing every quant desk eventually gets burned by: the Wald interval, for all its convenience, can silently fail exactly where it matters most, near a parameter boundary.

Learning objectives. By the end of this note you should be able to: