Confidence Intervals for Known Distributions and Pivotal Values 🌱

Last updated on August 10, 2020

A $1 - α$ confidence interval for $θ$ is an interval $(L, U)$ where we have the following:

L = g_{L} (X_{1}, \dots, X_{n}) and U = g_{U} (X_{1}, \dots, X_{n}) such that P (θ \in (L, U)) \geq 1 - α

The confidence interval itself is random, since $L$ and $U$ are functions of random variables
The parameter $θ$ is not random.
A confidence interval is a probability statement that a random interval captures a fixed parameter.

A random variable $V = g (X_{1}, \dots, X_{n}, θ)$ is a pivotal quantity if its distribution does not depend on the unknown parameter $θ$

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

A credible interval is the Bayesian equivalent of the confidence interval where the estimated interval is dependent on the prior...

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