# Unbiased hypothesis tests

• Is there some textbook or expository account showing that the definition of "unbiased test" bears the same sort of relation to "unbiased estimator" that interval estimation generally bears to confidence intervals? (One rejects $H_0 : \mu=\text{whatever}$ precisely if $\text{whatever}$ is not within a confidence interval for $\mu$.)
• Similarly, an account of the relationship between the concept of "uniformly most powerful test" and that of "uniformly most accurate confidence interval"?
• If I'm not mistaken, the F-test of $\sigma_1^2=\sigma_2^2$ versus "$\ne$", where these are variances of normally distributed populations, is not unbiased. Does that actually make it an inferior test, and what improvements on it are thereby called for?
• This, and much more, is covered in the classical text E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses, 3rd. ed., Springer. See, in particular, Chapter 5, Unbiasedness: Applications to Normal Distributions, especially Sections 5.3, Comparing the Means and Variances of Two Normal Distributions, Section 5.4, Confidence Intervals and Families of Tests and Section 5.5, Unbiased Confidence Sets. (cont.) – cardinal Apr 4 '13 at 2:23
• If you are unfamiliar with the text (which would surprise me), you may need to fill in gaps in notation by looking at Chapters 3 and 4. This reference shows that, indeed, the $F$-test you reference is UMP unbiased and, by duality, yields most accurate unbiased confidence intervals for the ratio $\sigma_1^2/\sigma_2^2$. – cardinal Apr 4 '13 at 2:26
• I have Lehmann's Theory of Point Estimation on a shelf next to me. His hypothesis testing book of course I know of, but I'm less familiar with it than with the estimation book. Let's see, looking at my question now, I can see what may be two instances of haste in phrasing. Maybe I'll edit it further tomorrow..... – Michael Hardy Apr 4 '13 at 4:36