Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.

翻译：分类优化因其实际重要性，在过去几十年中得到了积极探索。尽管已有大量文献涉及优化算法和潜在评分估计，但最优分类的不确定性量化仍有待探索，且具有重要的实际意义。决策者可能无需估计和恢复完整的最优选择集，而只需检验最优分类是否具有某一特定属性，例如最优分类是否应包含若干感兴趣的产品，或最优分类应包含多少个产品类别。本文提出了一种用于检验此类属性的新型推断框架。我们采用广泛使用的多项逻辑（MNL）模型，假设每位顾客会以与产品内在偏好评分成比例的概率，从所提供产品中选择一项。我们将对一般最优分类属性的推断，简化为对边际收益差距符号变化点检测的不确定性量化。我们证明了边际收益差距估计量的渐近正态性，并利用差距估计量构造了一个最大统计量来检测符号变化点。通过采用乘子自举技术近似最大统计量的分布，我们提出了一种有效的检验方法。我们还进行了数值实验来评估所提方法的性能。