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We study 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. This policy is inherently challenging to analyze because the recursion induces propagation of the estimation error backwards in time. In this work, 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. To explain, 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. 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.
We study the existence, strong consistency and asymptotic normality of estimators obtained from estimating functions, that are p-dimensional martingale transforms. The problem is motivated by the analysis of evolutionary clustered data, with distribu
Let $X$ be a mean zero Gaussian random vector in a separable Hilbert space ${mathbb H}$ with covariance operator $Sigma:={mathbb E}(Xotimes X).$ Let $Sigma=sum_{rgeq 1}mu_r P_r$ be the spectral decomposition of $Sigma$ with distinct eigenvalues $mu_1
We consider the canonical periodic review lost sales inventory system with positive lead-times and stochastic i.i.d. demand under the average cost criterion. We introduce a new policy that places orders such that the expected inventory level at the t
Classification rules can be severely affected by the presence of disturbing observations in the training sample. Looking for an optimal classifier with such data may lead to unnecessarily complex rules. So, simpler effective classification rules coul
Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, are an important special case. We consider an intrinsic