Trouble Understanding the Formal Definition of a Confidence IntervalA 1−α 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 thatPθ(θ∈Cn)&gt;1−α, for all θ∈ΘIf θ is a vector then we use a confidence set (such as a sphere or an ellipse) instead of an interval.Question: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(θ∈Cn). What is the sample space which P is drawing from? What is the set θ∈Cn? It seems here θ is being treated both as a fixed value (from the notation Pθ) and as a random variable (by the notation θ∈Cn).

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Trouble Understanding the Formal Definition of a Confidence IntervalA 1−α 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 thatPθ(θ∈Cn)&amp;gt;1−α, for all  θ∈ΘIf θ is a vector then we use a confidence set (such as a sphere or an ellipse) instead of an interval.Question: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&#039;t understand how this formality is capturing this concept.In particular, I don&#039;t understand what is meant by the notion of Pθ(θ∈Cn). What is the sample space which P is drawing from? What is the set θ∈Cn? It seems here θ is being treated both as a fixed value (from the notation Pθ) and as a random variable (by the notation θ∈Cn).

vadomosigmx6 · Accepted Answer

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&#039;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 {θ∈Cn} is shorthand for{ω:a(X1(ω),X2(ω),…,Xn(ω))&amp;lt;θ&amp;lt;b(X1(ω),X2(ω),…,Xn(ω))}tag1Viewed this way, the event (1) is more a statement about the random endpoints a and b, than about the parameter θ: it&#039;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&#039;s not what this definition appears to be about.

Trouble Understanding the Formal Definition of a Confidence

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