Technical Note—Data-Driven Profit Estimation Error in the Newsvendor Model

In this note, we identify a statistically significant error in naively estimating the expected profit in a data-driven newsvendor model, and we show how to correct the error. In particular, we analyze a newsvendor model where the continuous demand distribution is not known, and only a sample of demand data is available. In this context, an empirical demand distribution, that is induced by the sample of data, is used in place of the (unknown) true distribution. The quantity at the critical percentile $1-c/p$ of the empirical distribution is known as the sample average approximation order quantity, where p is the unit revenue and c the unit cost. We prove that, if the empirical distribution is used to estimate the expected profit, this estimate exhibits a positive, statistically significant bias. We derive a closed-form expression for this bias that only depends on p and c and the sample of data. The bias expression can then be used to design an adjusted expected profit estimate, which we prove is asymptotically unbiased. Numerical hypothesis testing experiments confirm that the unadjusted estimation error is statistically significant, whereas the adjusted estimation error is not significantly different from zero. The bias is not negligible in our numerical experiments: For lognormally and normally distributed demand, the unadjusted error is 2.4% and 3.0% of the true expected profit, respectively. A more detailed exploration with exact finite-sample results for exponentially distributed demand demonstrates that the estimation error percentage can be much larger.

Funding: A. F. Siegel gratefully acknowledges the support of the Grant I. Butterbaugh Professorship. M. R. Wagner gratefully acknowledges the support of a Neal and Jan Dempsey Faculty Fellowship. Both are from the Michael G. Foster School of Business, University of Washington.

Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2023.0070.