Estimating the size of a set using cascading exclusion

Let S be a finite set, and X1,ldots,Xn an i.i.d. uniform sample from S. To estimate the size |S|, without further structure, one can wait for repeats and use the birthday problem. This requires a sample size of the order |S|^frac12. On the other hand, if S=1,2,ldots,|S|, the maximum of the sample blown up by n/(n-1) gives an efficient estimator based on any growing sample size. This paper gives refinements that interpolate between these extremes. A general non-asymptotic theory is developed. This includes estimating the volume of a compact convex set, the unseen species problem, and a host of testing problems that follow from the question `Is this new observation a typical pick from a large prespecified population?' We also treat regression style predictors. A general theorem gives non-parametric finite n error bounds in all cases.