*Bounty: 100*

*Bounty: 100*

I was thinking about the formal definition of confidence intervals.

Given a Random Sample $textbf{X} = X_1,X_2,dots,X_n$ a level $alpha$ confidence interval for the population parameter $theta$ is defined as a pair of estimators (function of $textbf{X}$), namely $L(textbf{X})$ and $U(textbf{X})$ with $L(textbf{X}) leq U(textbf{X})$, with the property:

$$P(L(textbf{X}) leq theta leq U(textbf{X})) = 1-alpha$$

My interpretation of this this last equality is the joint probability of $L(textbf{X})$ and $U(textbf{X})$, i.e.

$$P(L(textbf{X}) leq theta, U(textbf{X})geqtheta) = 1-alpha$$

My question is how to work with this expression to find $L(textbf{X})$ and $U(textbf{X})$?

If I call $f_{L,U}(l,u)$ the joint pdf of $L(textbf{X})$ and $U(textbf{X})$ it should be something like:

$$int_{- infty}^{theta}int_{theta}^{+infty}f_{L,U}(l,u)dl du = 1 – alpha$$

and then I am stucked. I don’t know how to go on.

**My first question is**: since both $L(textbf{X})$ and $U(textbf{X})$ are function of the same $textbf{X}$ does something like the joint density even make sense?

I don’t know if my calculation is right but I found that something similar to a joint CDF for $L(textbf{X})$ and $U(textbf{X})$ could be (if $X in rm {I!R} $)

$$F_{L,U}(l,u)= F_{textbf{X}}bigg(minbig(L^{-1}(l),U^{-1}(u)big)bigg)$$

if that makes any sense at all. What is the correct way to think about this?

$$———————————–$$

I know that there are methods like **pivotal quantities** but the difference with respect to my case is that when we have Pivot the probabilistic statement is expressed it terms of **only one random variable** so I don’t have a joint density.

Suppose that I call $Q(textbf{X},theta)$ my pivot, I can find say $l$ and $u$ such that:

$$Pbig(lleq Q(textbf{X},theta) leq u big) = 1 – alpha$$

and then I can invert this relationship wrt $theta$. I suppose this equality could be rewritten as:

$$F_{Q}(u) – F_{Q}(l) = 1 – alpha$$

This last equation has infinite solutions since I have 2 unknowns $l$ and $u$. My understanding is that, in order to solve it, one has to choose value for one between $l$ and $u$. For instance I could choose $l$ such that $F_{Q}(l) = 2%$ and solve fo $u$ so that $F_{Q}(u) = 1 – alpha + 2%$

**And here is my second question:** so there the $level- alpha$ confidence intervals are infinite? And the difference between them is their length? Is my reasoning correct?

Any help would be much appreciated! Thank you.