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"Statistical estimation"
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Dynamic Pricing with Multiple Products and Partially Specified Demand Distribution
We study a dynamic pricing problem with multiple products and infinite inventories. The demand for these products depends on the selling prices and on parameters unknown to the seller. Their value can be learned from accumulating sales data using statistical estimation techniques. The quality of the parameter estimates is influenced by the amount of price dispersion; however, a large amount of variation in the selling prices can be costly since it means that suboptimal prices are used. The seller thus needs to balance optimizing the quality of the parameter estimates and optimizing instant revenue, i.e., exploitation and exploration.
In this study we propose a pricing policy for this dynamic pricing problem. The key idea is to use at each time period the price that is optimal with respect to current parameter estimates, with an additional constraint that ensures sufficient price dispersion. We measure the price dispersion by the smallest eigenvalue of the design matrix and show how a desired growth rate of this eigenvalue can be achieved by a simple quadratic constraint in the price-optimization problem. We study the performance of our pricing policy by providing bounds on the regret, which measures the expected revenue loss caused by using suboptimal prices.
Journal Article
A Nonparametric Approach to Multiproduct Pricing
Developed by General Motors (GM), the Auto Choice Advisor website ( http://www.autochoiceadvisor.com ) recommends vehicles to consumers based on their requirements and budget constraints. Through the website, GM has access to large quantities of data that reflect consumer preferences. Motivated by the availability of such data, we formulate a nonparametric approach to multiproduct pricing.
We consider a class of models of consumer purchasing behavior, each of which relates observed data on a consumers requirements and budget constraint to subsequent purchasing tendencies. To price products, we aim at optimizing prices with respect to a sample of consumer data. We offer a bound on the sample size required for the resulting prices to be near-optimal with respect to the true distribution of consumers. The bound exhibits a dependence of O(n log n) on the number n of products being priced, showing thatin terms of sample complexitythe approach is scalable to large numbers of products.
With regards to computational complexity, we establish that computing optimal prices with respect to a sample of consumer data is NP-complete in the strong sense. However, when prices are constrained by a price ladderan ordering of prices defined prior to price determinationthe problem becomes one of maximizing a supermodular function with real-valued variables. It is not yet known whether this problem is NP-hard. We provide a heuristic for our price-ladder-constrained problem, together with encouraging computational results.
Finally, we apply our approach to a data set from the Auto Choice Advisor website. Our analysis provides insights into the current pricing policy at GM and suggests enhancements that may lead to a more effective pricing strategy.
Journal Article
Semiparametric Approach to Dimension Reduction
We provide a novel and completely different approach to dimension-reduction problems from the existing literature. We cast the dimension-reduction problem in a semiparametric estimation framework and derive estimating equations. Viewing this problem from the new angle allows us to derive a rich class of estimators, and obtain the classical dimension reduction techniques as special cases in this class. The semiparametric approach also reveals that in the inverse regression context while keeping the estimation structure intact, the common assumption of linearity and/or constant variance on the covariates can be removed at the cost of performing additional nonparametric regression. The semiparametric estimators without these common assumptions are illustrated through simulation studies and a real data example. This article has online supplementary material.
Journal Article
A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation
This article proposes a constrained ℓ
1
minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p-variate random variables. The resulting estimator is shown to have a number of desirable properties. In particular, the rate of convergence between the estimator and the true s-sparse precision matrix under the spectral norm is
when the population distribution has either exponential-type tails or polynomial-type tails. We present convergence rates under the elementwise ℓ
∞
norm and Frobenius norm. In addition, we consider graphical model selection. The procedure is easily implemented by linear programming. Numerical performance of the estimator is investigated using both simulated and real data. In particular, the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods.
Journal Article
Bias-Corrected Matching Estimators for Average Treatment Effects
In Abadie and Imbens (2006), it was shown that simple nearest-neighbor matching estimators include a conditional bias term that converges to zero at a rate that may be slower than N
1/2
. As a result, matching estimators are not N
1/2
-consistent in general. In this article, we propose a bias correction that renders matching estimators N
1/2
-consistent and asymptotically normal. To demonstrate the methods proposed in this article, we apply them to the National Supported Work (NSW) data, originally analyzed in Lalonde (1986). We also carry out a small simulation study based on the NSW example. In this simulation study, a simple implementation of the bias-corrected matching estimator performs well compared to both simple matching estimators and to regression estimators in terms of bias, root-mean-squared-error, and coverage rates. Software to compute the estimators proposed in this article is available on the authors' web pages (
http://www.economics.harvard.edu/faculty/imbens/software.html
) and documented in Abadie et al. (2003).
Journal Article
Robust Inference With Multiway Clustering
In this article we propose a variance estimator for the OLS estimator as well as for nonlinear estimators such as logit, probit, and GMM. This variance estimator enables cluster-robust inference when there is two-way or multiway clustering that is nonnested. The variance estimator extends the standard cluster-robust variance estimator or sandwich estimator for one-way clustering (e.g., Liang and Zeger 1986; Arellano 1987) and relies on similar relatively weak distributional assumptions. Our method is easily implemented in statistical packages, such as Stata and SAS, that already offer cluster-robust standard errors when there is one-way clustering. The method is demonstrated by a Monte Carlo analysis for a two-way random effects model; a Monte Carlo analysis of a placebo law that extends the state-year effects example of Bertrand, Duflo, and Mullainathan (2004) to two dimensions; and by application to studies in the empirical literature where two-way clustering is present.
Journal Article
The Global Prevalence of Intimate Partner Violence Against Women
Data from 81 countries was used to estimate global prevalence of intimate partner violence against women. Violence against women is a phenomenon that persists in all countries ( 1 ). Since the 1993 World Conference on Human Rights and the Declaration on the Elimination of Violence against Women, the international community has acknowledged that violence against women is an important public health, social policy, and human rights concern. However, documenting the magnitude of violence against women and producing reliable comparative data to guide policy and monitor progress has been difficult.
Journal Article
Seasonal Incidence of Symptomatic Influenza in the United States
The seasonal incidence of influenza is often approximated as 5%-20%.
We used 2 methods to estimate the seasonal incidence of symptomatic influenza in the United States. First, we made a statistical estimate extrapolated from influenza-associated hospitalization rates for 2010-2011 to 2015-2016, collected as part of national surveillance, covering approximately 9% of the United States, and including the existing mix of vaccinated and unvaccinated persons. Second, we performed a literature search and meta-analysis of published manuscripts that followed cohorts of subjects during 1996-2016 to detect laboratory-confirmed symptomatic influenza among unvaccinated persons; we adjusted this result to the US median vaccination coverage and effectiveness during 2010-2016.
The statistical estimate of influenza incidence among all ages ranged from 3.0%-11.3% among seasons, with median values of 8.3% (95% confidence interval [CI], 7.3%-9.7%) for all ages, 9.3% (95% CI, 8.2%-11.1%) for children <18 years, and 8.9% (95% CI, 8.2%-9.9%) for adults 18-64 years. Corresponding values for the meta-analysis were 7.1% (95% CI, 6.1%-8.1%) for all ages, 8.7% (95% CI, 6.6%-10.5%) for children, and 5.1% (95% CI, 3.6%-6.6%) for adults.
The 2 approaches produced comparable results for children and persons of all ages. The statistical estimates are more versatile and permit estimation of season-to-season variation. During 2010-2016, the incidence of symptomatic influenza among vaccinated and unvaccinated US residents, including both medically attended and nonattended infections, was approximately 8% and varied from 3% to 11% among seasons.
Journal Article
Bayesian Nonparametric Modeling for Causal Inference
Researchers have long struggled to identify causal effects in nonexperimental settings. Many recently proposed strategies assume ignorability of the treatment assignment mechanism and require fitting two models-one for the assignment mechanism and one for the response surface. This article proposes a strategy that instead focuses on very flexibly modeling just the response surface using a Bayesian nonparametric modeling procedure, Bayesian Additive Regression Trees (BART). BART has several advantages: it is far simpler to use than many recent competitors, requires less guesswork in model fitting, handles a large number of predictors, yields coherent uncertainty intervals, and fluidly handles continuous treatment variables and missing data for the outcome variable. BART also naturally identifies heterogeneous treatment effects. BART produces more accurate estimates of average treatment effects compared to propensity score matching, propensity-weighted estimators, and regression adjustment in the nonlinear simulation situations examined. Further, it is highly competitive in linear settings with the \"correct\" model, linear regression. Supplemental materials including code and data to replicate simulations and examples from the article as well as methods for population inference are available online.
Journal Article