Accurate confidence intervals for distributions with one parameter

Summary

Let$\hat \theta _n$ be an estimate of a real parameter θ. Suppose that for some functionc(·) and some random variable (r.v.) τn, the distribution of

$$Z_n = \left( {\left( \theta \right) - c\left( {\hat \theta _n } \right)} \right)/\tau _n $$

is continuous and depends only on θ andn and that the cumulants ofZ _n can be expanded in the form

$$K_r Z_n \approx \sum\limits_{i = r - 1}^\infty {a_{ri} (\theta )n^{ - i} } $$

.

Then a confidence interval for θ can be constructed with level 1−α+O(n^−j/2) for any given value of α andj.