Confidence & Credible Interval Estimates

Point estimates of unknown parameters θ governing the distribution of an observed quantity X are unsatisfying if they come with no measure of accuracy or precision. One approach to giving such a measure is to offer interval estimates for θ, rather than point estimates; upon observing X, we construct an interval [a, b] which is very likely to contain θ, and which is very short. The approach to exactly how these are constructed and interpreted is different for inference in the Sampling Theory tradition, and in the Bayesian tradition. In these notes I’ll present both approaches to estimating the means of the Normal and Exponential distributions, using “pivotal quantities,” and of Poisson random variables, using detailed features of the distribution, on the basis of a random sample of fixed size n.