Find Exact Confidence Intervals: Tips & Techniques

[logarithmic]

Does anyone know how to find exact confidence intervals? I've looked through textbooks, but they only find approximate CIs using the assumption that $$\frac{\hat{\theta}-\theta}{se(\hat\theta)}}\rightarrow Z.$$

So given a estimator, $$\hat\theta$$ do I have to find an exact distrubution for the above expression first. And is there any nice way to do this?

 

[ssd]

$$\frac{\hat{\theta}-\theta}{se(\hat\theta)}}\rightarrow Z.$$

This result is by CLT depending on certain conditions.
Of course exact CI's are available. This depends on distribution of the statistic. Example:
x1,x2,...,xn is a sample from N(mu,sigma). Sigma known. Exact CI for mu can be easily found (available in most of textbooks of appropriate standard).

 

[logarithmic]

This result is by CLT depending on certain conditions.
Of course exact CI's are available. This depends on distribution of the statistic. Example:
x1,x2,...,xn is a sample from N(mu,sigma). Sigma known. Exact CI for mu can be easily found (available in most of textbooks of appropriate standard).

Yeah, I'm aware of that. Your example relies on the fact that you know exactly the distribution of the expression above, which is normal. But what if you can't find that easily, e.g. if your X_i's are from an exponential distribution.

 

[ssd]

Yeah, I'm aware of that. Your example relies on the fact that you know exactly the distribution of the expression above, which is normal. But what if you can't find that easily, e.g. if your X_i's are from an exponential distribution.

Your question does not appear very specific to me. Of course one needs to know the distribution of the statistic. There is no unique or universal way to find distributions of all statistics from all distributions. If the distribution cannot be enumerated then one tries large sample approximations or numerical simulation.