Error Propagation in Asymptotic Analysis of the Data-Driven (s, S) Inventory Policy

We study periodic review stochastic inventory control in the data-driven setting where the retailer makes ordering decisions based only on historical demand observations without any knowledge of the probability distribution of the demand. Because an (s, S)-policy is optimal when the demand distribution is known, we investigate the statistical properties of the data-driven (s, S)-policy obtained by recursively computing the empirical cost-to-go functions. This policy is inherently challenging to analyze because the recursion induces propagation of the estimation error backward in time. In this work, we establish the asymptotic properties of this data-driven policy by fully accounting for the error propagation. In this setting, the empirical cost-to-go functions for the estimated parameters are not i.i.d. sums because of the error propagation. Our main methodological innovation comes from an asymptotic representation for multi-sample U-processes in terms of i.i.d. sums. This representation enables us to apply empirical process theory to derive the influence functions of the estimated parameters and to establish joint asymptotic normality. Based on these results, we also propose an entirely data-driven estimator of the optimal expected cost, and we derive its asymptotic distribution. We demonstrate some useful applications of our asymptotic results, including sample size determination and interval estimation.

Funding: This work was supported by Singapore MOE AcRF Tier 2 [A-8001052-00-00] and the National Natural Science Foundation of China [72071138].

Supplemental Material: The computer code and data that support the findings of this study are available within this article’s supplemental material at https://doi.org/10.1287/opre.2020.0568.