say you have an estimator that is consistent, but biased. even if asymtpotically normal (clt) and all, how does hypothesis testing make sense?
to form the t statistic, from what i understand, you either have normally distributed data, or n large enough and the clt to hold so that the estimator is asymptotically normal. then, after establishing asymptotic normality, subtracting the mean, and dividing by the standard error gives you an approximate z -distribution, and then dividing by another particular z- RV yields the t statistic.
It seems to me that just relying on a consistent estimator, subtracting the mean to get the z, does not apply as the mean of the estimtor is not necessarily the true parameter of interest, given that the estimator is biased. Is the justification that the estiamtor is roughyl unbiased in an infinite sample?