ESTIMATION AND EVALUATION OF DATA-DRIVEN INVENTORY POLICIES BY ASYMPTOTIC STATISTICS

Abstract

This thesis studies periodic review stochastic inventory control in the data-driven setting, in which the retailer makes ordering decisions based only on historical demand observations without any knowledge of the probability distribution of the demand. Since 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 (called EDP estimator). This estimator is inherently challenging to analyze because the recursion induces propagation of the estimation error backwards in time. In this thesis, we establish the asymptotic properties of this data-driven policy by fully accounting for the error propagation. First, we rigorously show the consistency of the estimated parameters by filling in some gaps (due to unaccounted error propagation) in the existing studies. On the other hand, empirical process theory cannot be directly applied to show asymptotic normality since the empirical cost-to-go functions for the estimated parameters are not i.i.d. sums, again due to 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 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. Beyond deriving the asymptotic distribution of our EDP estimators, we further investigate the semiparametric efficiency of the proposed estimators. We show that the asymptotic variances of EDP estimators match the statistical lower bound and so the proposed estimators are asymptotically efficient. The extensions to dependent demand are also investigated in this thesis, where we propose a SAA type estimator to estimate the optimal expected cost under base stock polices. We demonstrate some useful applications of our asymptotic results, including sample size determination, as well as interval estimation and hypothesis testing on vital parameters of the inventory problem. The results from our numerical simulations conform to our theoretical analysis.