We study a pricing problem with finite inventory and semi-parametric demand uncertainty. Demand is a price-dependent Poisson process whose mean is the product of buyers’ arrival rate, which is a constant \(\lambda \), and buyers’ purchase probability \(q(p)\), where p is the price. The seller observes arrivals and sales, and knows neither \(\lambda \) nor \(q\). Based on a non-parametric maximum-likelihood estimator of \((\lambda ,q)\), we construct an estimator of mean demand and show that as the system size and number of prices grow, it is asymptotically more efficient than the maximum likelihood estimator based only on sale data. Based on this estimator, we develop a pricing algorithm paralleling (Besbes and Zeevi in Oper Res 57:1407–1420, 2009) and study its performance in an asymptotic regime similar to theirs: the initial inventory and the arrival rate grow proportionally to a scale parameter n. If \(q\) and its inverse function are Lipschitz continuous, then the worst-case regret is shown to be \(O((\log n / n)^{1/4})\). A second model considered is the one in Besbes and Zeevi (2009, Section 4.2), where no arrivals are involved; we modify their algorithm and improve the worst-case regret to \(O((\log n / n)^{1/4})\). In each setting, the regret order is the best known, and is obtained by refining their proof methods. We also prove an \(\Omega (n^{-1/2})\) lower bound on the regret. Numerical comparisons to state-of-the-art alternatives indicate the effectiveness of our arrivals-based approach.

我们研究具有有限库存和半参数需求不确定性的定价问题。需求是一个与价格有关的泊松过程，其均值是买方到达率（是常数\（\ lambda \）和买方购买概率\（q（p）\））的乘积，其中p是价格。卖方观察到货和销售，不知道\（\ lambda \）也不知道\（q \）。基于\（（\ lambda，q）\）的非参数最大似然估计，我们构造了一个平均需求估计量，并表明随着系统规模和价格数量的增长，它比仅基于销售数据的最大似然估计量渐近更有效。基于此估算器，我们开发了一种并行的定价算法（Besbes和Zeevi，Oper Res 57：1407–1420，2009），并研究了其在类似于它们的渐近状态下的性能：初始库存和到达率成比例增长参数n。如果\（q \）及其反函数是Lipschitz连续的，则最坏情况的后悔是\（O（（\ log n / n）^ {1/4}）\）。考虑的第二种模型是Besbes和Zeevi（2009，第4.2节）中没有涉及到的模型。我们修改了他们的算法，并将最坏情况的遗憾改善为\（O（（\ log n / n）^ {1/4}）\）。在每种情况下，后悔顺序是众所周知的，并且是通过改进其证明方法而获得的。我们还证明了遗憾的\（\ Omega（n ^ {-1/2}）\）下界。与最先进的替代方案的数值比较表明了我们基于到达的方法的有效性。