Trouble Understanding the Formal Definition of a Confidence

Kendal Day

Kendal Day

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Trouble Understanding the Formal Definition of a Confidence Interval
A 1α confidence interval for a paramater θ is an interval Cn=(a,b) where a=a(X1,,Xn) and b=b(X1,,Xn) are functions of the data such that
Pθ(θCn)>1α, for all  θΘ
If θ is a vector then we use a confidence set (such as a sphere or an ellipse) instead of an interval.
While I understand conceptually what a confidence interval is (i.e., a 95% CI means that 95% of experiments will trap the paramater in the interval), I don't understand how this formality is capturing this concept.
In particular, I don't understand what is meant by the notion of Pθ(θCn). What is the sample space which P is drawing from? What is the set θCn? It seems here θ is being treated both as a fixed value (from the notation Pθ) and as a random variable (by the notation θCn).

Answer & Explanation



Beginner2022-04-09Added 12 answers

Cn is an interval with random endpoints, denoted a and b. Both the endpoints are functions of your sample X1,X2,,Xn, and the joint distribution of the X's is parametrized by θ, hence the subscript on Pθ. The parameter θ that governs this joint distribution is nonrandom, and generally unknown (and the mission of the CI is to capture this unknown parameter). The set {θCn} is shorthand for
Viewed this way, the event (1) is more a statement about the random endpoints a and b, than about the parameter θ: it's asking whether the random left endpoint is less than the number θ and the random right endpoint is greater than the number θ. In the frequentist treatment of confidence intervals, θ is not a random variable; in other treatments it is possible to regard θ as the observed value of a random variable, but that's not what this definition appears to be about.

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