Pivotal Quantities and Confidence Intervals

A pivotal quantity has a distribution that does not depend on the unknown parameter, which lets one turn a probability statement into a confidence interval.

Definition

A pivotal quantity is a function of the data and the parameter whose probability distribution is the same for every parameter value; inverting a probability statement about the pivot yields a confidence interval for the parameter.

Scope

This topic covers the definition of a pivotal quantity, the pivotal method for building exact confidence intervals, canonical pivots in location-scale and normal models such as the t and chi-squared pivots, the choice of interval endpoints to control length and symmetry, and large-sample approximate pivots that give Wald-type intervals from asymptotic normality.

Core questions

• What distinguishes a pivot from an ordinary statistic, and why is parameter-free distribution essential?
• How does the pivotal method convert a probability statement into an interval?
• What are the standard pivots for the mean and variance of a normal sample?
• How do asymptotic pivots based on normality give approximate intervals when exact pivots are unavailable?

Key theories

Pivotal method

If a pivot has a known distribution, choosing quantiles that capture a given probability and solving the resulting inequalities for the parameter produces a confidence interval with exactly that coverage.

Asymptotic pivots and Wald intervals

When no exact pivot exists, an estimator minus the parameter divided by its standard error is approximately standard normal in large samples, yielding the familiar estimate-plus-or-minus-margin confidence interval.

Clinical relevance

The pivotal method produces the t-interval for a mean and the chi-squared interval for a variance that are reported throughout applied research, while asymptotic pivots give the estimate-plus-or-minus-margin intervals used for proportions, regression coefficients, and survey estimates.

History

Gosset's 1908 derivation of the t distribution under the pen name Student provided the first exact pivot for the normal mean, and Neyman's 1937 confidence theory placed the pivotal construction within a general frequentist framework.

Key figures

• Jerzy Neyman
• William Sealy Gosset
• Ronald A. Fisher
• George Casella

Seminal works

• casella2002

Frequently asked questions

What makes a quantity pivotal?

Its distribution must be exactly the same for every value of the unknown parameter; only then can quantiles be chosen without knowing the parameter, which is what allows an interval with guaranteed coverage.

Are Wald intervals exact?

No. They rely on the asymptotic normality of the estimator and so have only approximate coverage in finite samples, which can be poor for small samples or parameters near a boundary such as a proportion close to zero or one.