Lecture 12 slides: Confidence sets

So far, we have been considering point estimation. In this lecture, we will study interval estimation. Let X denote our data. Let θ ∈ R be our parameter of interest. Our task is to construct a data-dependent interval [l(X), r(X)] so that it contains θ with large probability. One possibility is to set l(X) = −∞ and r(X) = +∞. Such an interval will contain θ with probabity 1. Of course, the problem with this interval is that it is too long. So, we want to construct an interval as short as possible. More generally, instead of intervals, we can consider con dence set C(X) ⊂ R such that it contains θ with large probability and has as small volume as possible. The concept of con dence sets can be also applied for any set of possible parameter values Θ, not just for R. Let us introduce basic concepts related to con dence sets.