Assuming that a sample estimator converges to some unknown constant (a wild assumption to be sure) and without assuming the distribution of either the sample estimator or the variables from which it was calculated, is there a general foolproof way to construct a confidence interval for that unknown constant?

Perusing some nonparametric statistics lectures online, the most widely described "general method" involves bootstrapping. On the other hand, bootstrapping doesn't work for some estimators e.g. estimating $\theta$ for $U(0, \theta)$, and the material on the conditions where bootstrapping works is hard to grasp for me. I've also come across subsampling, but while it's purportedly more widely applicable than bootstrapping and just as easy to code, I still don't understand when it's applicable anymore than I know for bootstrapping. It'd be great if more experienced people can point me in the right direction.

  • $\begingroup$ What do you mean unknown constant? How are you estimating this constant? Is it a mean? $\endgroup$ – user2974951 Jan 23 at 9:55
  • $\begingroup$ In search of a general method, I'm intentionally not specifying what the sample estimator or the constant to which it converges is. The sample estimator would be computed from a random sample. If the sample estimator is the sample mean, the constant would indeed be the expected value I'm assuming to exist for the unknown distribution the sample is drawn from. $\endgroup$ – BatWannaBe Jan 23 at 10:01
  • $\begingroup$ If we are talking really generally here then bootstrap is probably the best option, since it makes no assumptions about the data. I have not heard of subsampling being used for estimating anything. $\endgroup$ – user2974951 Jan 23 at 15:01
  • $\begingroup$ Oh well, bootstrap it is then. As for subsampling, I've seen it said to work in some cases the bootstrap doesn't, and some go as far as to say it's generally more applicable if not for the pesky detail of choosing a good resample size. $\endgroup$ – BatWannaBe Jan 23 at 16:48

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