Confidence Intervals for Known Distributions and Pivotal Values 🌱

A \(1-\alpha\) confidence interval for \(\theta\) is an interval \((L,U)\) where we have the following:

\[L=g_{L}\left(X_{1}, \ldots, X_{n}\right) \text{ and } U=g_{U}\left(X_{1}, \ldots, X_{n}\right) \text{ such that } P(\theta \in(L, U)) \geq 1-\alpha\]
  • The confidence interval itself is random, since $L$ and $U$ are functions of random variables
  • The parameter \(\theta\) is not random.
  • A confidence interval is a probability statement that a random interval captures a fixed parameter.

Pivotal quantity

A random variable \(V = g(X_1,\ldots, X_n,\theta )\) is a pivotal quantity if its distribution does not depend on the unknown parameter \(\theta\)

Finding Confidence Intervals

  1. Find a pivotal quantity \(V\)
  2. Choose \(a\) and \(b\) such that \(P(a<V<b) \geq 1-\alpha\)
    • This can be done by choosing \(a\) and \(b\) such that \(P(V<a) = P(V>b) = \frac{\alpha}{2}\)
  3. Could also do something very similar for a one-sided confidence interval (just need either \(a\) or \(b\))

Notes mentioning this note

What is a Credible Interval
A credible interval is the Bayesian equivalent of the confidence interval where the estimated interval is dependent on the prior...

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