Efficient Empirical Bayes Prediction Under Check Loss Using Asymptotic Risk Estimates

We develop a novel Empirical Bayes methodology for prediction under check loss in high-dimensional Gaussian models. The check loss is a piecewise linear loss function having differential weights for measuring the amount of underestimation or overestimation. Prediction under it differs in fundamental aspects from estimation or prediction under weighted-quadratic losses. Because of the nature of this loss, our inferential target is a pre-chosen quantile of the predictive distribution rather than the mean of the predictive distribution. We develop a new method for constructing uniformly efficient asymptotic risk estimates which are then minimized to produce effective linear shrinkage predictive rules. In calculating the magnitude and direction of shrinkage, our proposed predictive rules incorporate the asymmetric nature of the loss function and are shown to be asymptotically optimal. Using numerical experiments we compare the performance of our method with traditional Empirical Bayes procedures and obtain encouraging results

Empirical Bayes Prediction for the Multivariate Newsvendor Loss Function

We develop a novel Empirical Bayes methodology for prediction under check loss in high-dimensional Gaussian models. The check loss is a piecewise linear loss function having differential weights for measuring the amount of underestimation or overestimation. Prediction under it differs in fundamental aspects from estimation or prediction under weighted-quadratic losses. Because of the nature of this loss, our inferential target is a pre-chosen quantile of the predictive distribution rather than the mean of the predictive distribution. We develop a new method for constructing uniformly efficient asymptotic risk estimates which are then minimized to produce effective linear shrinkage predictive rules. In calculating the magnitude and direction of shrinkage, our proposed predictive rules incorporate the asymmetric nature of the loss function and are shown to be asymptotically optimal. Using numerical experiments we compare the performance of our method with traditional Empirical Bayes procedures and obtain encouraging results

The Value of Field Experiments

The feasibility of using field experiments to optimize marketing decisions remains relatively unstudied. We investigate category pricing decisions that require estimating a large matrix of cross-product demand elasticities and ask the following question: How many experiments are required as the number of products in the category grows? Our main result demonstrates that if the categories have a favorable structure, we can learn faster and reduce the number of experiments that are required: the number of experiments required may grow just logarithmically with the number of products. These findings potentially have important implications for the application of field experiments. Firms may be able to obtain meaningful estimates using a practically feasible number of experiments, even in categories with a large number of products. We also provide a relatively simple mechanism that firms can use to evaluate whether a category has a structure that makes it feasible to use field experiments to set prices. We illustrate how to accomplish this using either a sample of historical data or a pilot set of experiments. We also discuss how to evaluate whether field experiments can help optimize other marketing decisions, such as selecting which products to advertise or promote.National Science Foundation (U.S.) (Grant CMMI-0856063)National Science Foundation (U.S.) (Grant CMMI-1158658

Linearly Parameterized Bandits

We consider bandit problems involving a large (possibly infinite) collection of arms, in which the expected reward of each arm is a linear function of an $r$-dimensional random vector $\mathbf{Z} \in \mathbb{R}^r$, where $r \geq 2$. The objective is to minimize the cumulative regret and Bayes risk. When the set of arms corresponds to the unit sphere, we prove that the regret and Bayes risk is of order $\Theta(r \sqrt{T})$, by establishing a lower bound for an arbitrary policy, and showing that a matching upper bound is obtained through a policy that alternates between exploration and exploitation phases. The phase-based policy is also shown to be effective if the set of arms satisfies a strong convexity condition. For the case of a general set of arms, we describe a near-optimal policy whose regret and Bayes risk admit upper bounds of the form $O(r \sqrt{T} \log^{3/2} T)$.Comment: 40 pages; updated results and reference

The Assortment Packing Problem: Multiperiod Assortment Planning for Short-Lived Products

Motivated by retailers ’ frequent introduction of new items to refresh product lines and maintain their market shares, we present the assortment packing problem in which a firm must decide, in advance, the release date of each product in a given collection over a selling season. Our formulation models the trade-offs among profit margins, preference weights, and limited life cycles. A key aspect of the problem is that each product is short-lived in the sense that, once introduced, its attractiveness lasts only a few periods and vanishes over time. The objective is to determine when to introduce each product to maximize the total profit over the selling season. Even for two periods, the corresponding optimization problem is shown to be NP-complete. As a result, we study a continuous relaxation of the problem that approximates the problem well, when the number of products is large. When margins are identical and product preferences decay exponentially, its solution can be characterized: it is optimal to introduce products with slower decays earlier. The structural properties of the relaxation also help us to develop several heuristics, for which we establish performance guarantees. We test our heuristics with data on sales and release dates of woman handbags from an accessories retailer. The numerical experiments show that the heuristics perform very well and can yield significant improvements in profitability. 1

Identifying Early Buyers from Purchase Data

Market research has shown that consumers exhibit a variety of di#erent purchasing behaviors; specifically, some tend to purchase products earlier than other consumers. Identifying such early buyers can help personalize marketing strategies, potentially improving their e#ectiveness. In this paper, we present a non-parametric approach to the problem of identifying early buyers from purchase data. Our formulation takes as inputs the detailed purchase information of each consumer, with which we construct a weighted directed graph whose nodes correspond to consumers and whose edges correspond to purchases consumers have in common; the edge weights indicate how frequently consumers purchase products earlier than other consumers