ترغب بنشر مسار تعليمي؟ اضغط هنا

Asymptotic Analysis for Data-Driven Inventory Policies

95   0   0.0 ( 0 )
 نشر من قبل Xun Zhang
 تاريخ النشر 2020
  مجال البحث الاحصاء الرياضي
والبحث باللغة English




اسأل ChatGPT حول البحث

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.



قيم البحث

اقرأ أيضاً

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 tions belonging to the exponential family, and which may also vary in terms of other component series. Within a quasi-likelihood approach, we construct estimating equations, which accommodate different forms of dependency among the components of the response vector and establish multivariate extensions of results on linear and generalized linear models, with stochastic covariates. Furthermore, we characterize estimating functions which are asymptotically optimal, in that they lead to confidence regions for the regression parameters which are of minimum size, asymptotically. Results from a simulation study and an application to a real dataset are included.
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 $mu_1 >mu_2> dots$ and the corresponding spectral projectors $P_1, P_2, dots.$ Given a sample $X_1,dots, X_n$ of size $n$ of i.i.d. copies of $X,$ the sample covariance operator is defined as $hat Sigma_n := n^{-1}sum_{j=1}^n X_jotimes X_j.$ The main goal of principal component analysis is to estimate spectral projectors $P_1, P_2, dots$ by their empirical counterparts $hat P_1, hat P_2, dots$ properly defined in terms of spectral decomposition of the sample covariance operator $hat Sigma_n.$ The aim of this paper is to study asymptotic distributions of important statistics related to this problem, in particular, of statistic $|hat P_r-P_r|_2^2,$ where $|cdot|_2^2$ is the squared Hilbert--Schmidt norm. This is done in a high-complexity asymptotic framework in which the so called effective rank ${bf r}(Sigma):=frac{{rm tr}(Sigma)}{|Sigma|_{infty}}$ (${rm tr}(cdot)$ being the trace and $|cdot|_{infty}$ being the operator norm) of the true covariance $Sigma$ is becoming large simultaneously with the sample size $n,$ but ${bf r}(Sigma)=o(n)$ as $ntoinfty.$ In this setting, we prove that, in the case of one-dimensional spectral projector $P_r,$ the properly centered and normalized statistic $|hat P_r-P_r|_2^2$ with {it data-dependent} centering and normalization converges in distribution to a Cauchy type limit. The proofs of this and other related results rely on perturbation analysis and Gaussian concentration.
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 ime of arrival of an order is at a fixed level and call it the Projected Inventory Level (PIL) policy. We prove that this policy has a cost-rate superior to the equivalent system where excess demand is back-ordered instead of lost and is therefore asymptotically optimal as the cost of losing a sale approaches infinity under mild distributional assumptions. We further show that this policy dominates the constant order policy for any finite lead-time and is therefore asymptotically optimal as the lead-time approaches infinity for the case of exponentially distributed demand per period. Numerical results show this policy also performs superior relative to other policies.
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 d be achieved if we relax the goal of fitting a good rule for the whole training sample but only consider a fraction of the data. In this paper we introduce a new method based on trimming to produce classification rules with guaranteed performance on a significant fraction of the data. In particular, we provide an automatic way of determining the right trimming proportion and obtain in this setting oracle bounds for the classification error on the new data set.
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 principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Frechet mean function, and then performing a classical multivariate functional principal component analysis on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained with exponential maps. The tangent-space approximation through functional principal component analysis is shown to be well-behaved in terms of controlling the residual variation if the Riemannian manifold has nonnegative curvature. Specifically, we derive a central limit theorem for the mean function, as well as root-$n$ uniform convergence rates for other model components, including the covariance function, eigenfunctions, and functional principal component scores. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the sphere, illustrated with longitudinal fruit fly behavior patterns. RFPCA is shown to be superior in terms of trajectory recovery in comparison to an unrestricted functional principal component analysis in applications and simulations and is also found to produce principal component scores that are better predictors for classification compared to traditional functional functional principal component scores.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا