Pivotal Quantities and Exact Confidence Intervals

A point estimate reported without any sense of how far it might be from the truth is close to useless to a risk committee: "the fitted daily volatility is 1.8%" invites the obvious follow-up, how sure are you? Topic 3 built the estimators — method of moments, maximum likelihood — but stopped at the point estimate. This note builds the first, and best, kind of answer: an exact confidence interval, a random interval that is guaranteed, by construction and for every finite sample size, to trap the true parameter with a specified probability. The construction method is a single reusable trick — find a pivotal quantity, a function of the data and the parameter whose distribution does not depend on the parameter, then invert its distribution to isolate $\theta$. The $t$ and $\chi^2$ distributions built in The Chi-Squared, $t$, and $F$ Distributions turn out to be exactly the pivots needed for a normal mean and variance, so this note is largely a matter of putting already-built machinery to a new use — and, along the way, exposing a deep duality: every confidence interval is secretly a family of hypothesis tests in disguise, and vice versa.

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