New Insights into Sampling Without Replacement
Myatt & Ronayne: "Decreasing returns to sampling without replacement" Economics Letters, Volume 264 (2026)
David Ronayne (ESMT Berlin, Project A04) and David P. Myatt (London Business School) investigate the mathematical properties of sampling from a finite population without replacement, specifically focusing on how the expected values of order statistics evolve as the sample size increases. While it is a well-established statistical result that sampling from continuous populations yields decreasing returns (meaning the expected maximum draw is a discretely increasing and concave function of the sample size, and, that the expected minimum is a discretely decreasing and convex function of the sample size), this research confirms that the same logic applies to finite, discrete populations. This is particularly relevant for scenarios such as a buyer searching for the lowest price among a fixed set of competitors or an auctioneer determining how many bidders to invite to a sale.
The authors’ core result demonstrates that the expected sample minimum is weakly decreasing and discretely convex regardless of the specific population values. If the lowest value in the population is unique, the expected minimum decreases strictly as the sample size grows. Furthermore, if the second and third lowest values are distinct, the function is strictly discretely convex. These findings indicate that the marginal expected benefit of adding one more observation to a sample, when one cares about the extreme (lowest or highest) value of a sample, is always less than or equal to the benefit gained from the previous addition. For a consumer who cares about finding the lowest price, the value of retrieving an additional quotation falls with the total number of quotations retrieved.
Beyond extreme values, the researchers explore non-extreme order statistics, the kth lowest realisation in a sample. They identify a sufficient condition for these statistics to exhibit discrete convexity: the population must have “increasing spacings,” meaning the population values themselves form a discretely convex sequence. Under this condition, the expected value of the kth lowest observation is discretely convex in the sample size.
Link: Decreasing returns to sampling without replacement
