Estimating the size of a set using cascading exclusion
For statisticians and machine learning researchers, this work provides a unified theoretical framework with finite-sample guarantees for set size estimation across diverse domains.
This paper develops a general non-asymptotic theory for estimating the size of a finite set using cascading exclusion, interpolating between the birthday problem and maximum-based estimators. It provides finite-sample error bounds for applications including volume estimation, unseen species, and testing whether a new observation is typical from a prespecified population.
Let $S$ be a finite set, and $X_1,\ldots,X_n$ 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 $|S|^\frac{1}{2}$. 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.