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Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org Profit Estimation Error in the Newsvendor Model Under a Parametric Demand Distribution

Andrew F. Siegel , Michael R. Wagner

To cite this article: Andrew F. Siegel , Michael R. Wagner (2021) Profit Estimation Error in the Newsvendor Model Under a Parametric Demand Distribution. Management Science 67(8):4863-4879. https://doi.org/10.1287/mnsc.2020.3766

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Profit Estimation Error in the Newsvendor Model Under a Parametric Demand Distribution

Andrew F. Siegel,a,b Michael R. Wagnera a Information Systems and Operations Management, Michael G. Foster School of Business, University of Washington, Seattle, Washington 98195; b Finance and Business Economics, Department of Statistics, University of Washington, Seattle, Washington 98195 Contact: [email protected], https://orcid.org/0000-0002-2144-2591 (AFS); [email protected], https://orcid.org/0000-0003-2077-3564 (MRW)

Received: June 16, 2019 Abstract. We consider the newsvendor model in which uncertain demand is assumed to Revised: December 6, 2019; April 7, 2020; follow a probabilistic distribution with known functional form but unknown parameters. April 29, 2020 These parameters are estimated, unbiasedly and consistently, from data. We show that the May 17, 2020 Accepted: classic maximized expected profit expression exhibits a systematic expected estimation Published Online in Articles in Advance: December 28, 2020 error. We provide an asymptotic adjustment so that the estimate of maximized expected profit is unbiased. We also study expected estimation error in the optimal order quantity, https://doi.org/10.1287/mnsc.2020.3766 which depends on the distribution: (1) if demand is exponentially or normally distributed, the order quantity has zero expected estimation error; (2) if demand is log-normally Copyright: © 2020 INFORMS distributed, there is a nonzero expected estimation error in the order quantity that can be corrected. Numerical experiments, for light- and heavy-tailed distributions, confirm our theoretical results.

History: Accepted by Vishal Gaur, operations management. Funding: The first author gratefully acknowledges the support of the Grant I. Butterbaugh Profes- sorship. The second author gratefully acknowledges the support of a Neal and Jan Dempsey Faculty Fellowship. Supplemental Material: The online electronic companion is available at https://doi.org/10.1287/mnsc.2020.3766.

Keywords: newsvendor • estimation error • statistics • Fisher information

1. Introduction chain overall.” Academic research has also identified In this paper, we show that the classic newsvendor the costs of positive and negative bias: Cassar and model can systematically overestimate its maximized Gibson (2008)report,“Higher costs of obsolescence expected profit; in other words, the expected profit and inventory holding (Lee and Adam 1986,Watson perceived by a decision maker can be larger on average 1987) and lower returns on capital investments may than what actually occurs. In practice, the expected result from optimistic forecasts, while stock-out costs profitservesasaforecast for the future realized profit, and reputation damage are likely consequences of and we effectively show that this forecast is biased. pessimistic forecasts (Ittner and Larcker 1998,Durand The American Production and Inventory Control 2003).” Furthermore, Lipson (2019)isaUniversityof Society (APICS), a worldwide professional organiza- Virginia case study that teaches financial statement tionfoundedin1957withmorethan45,000members, forecasting and states that forecasts should be unbi- specifies that “a normal property of a good forecast is ased (neither optimistic nor pessimistic) because bi- that it is not biased” (APICS 2019). Similarly, the ased forecasts lead to biased decisions. More infor- Institute of Business Forecasting and Planning pro- mally, if a manager chronically overestimates profit, vides resources for measuring and correcting fore- eventually it will be noticed that the errors cannot casting bias (Singh 2017). IBM has also provided be solely attributed to forecasting error but rather guidance on correcting forecasting bias (Parks 2018), to mistaken calculations (i.e., forecasting bias). In which is summarized as “error isn’t always address- this paper, we provide an approach to correct these able but bias usually is. Consider focusing on bias first mistaken calculations in the context of the news- when trying to improve your forecasts.” An article vendor model and produce bias-adjusted forecasts discussing the correction of forecasting bias has also of profit. appeared on LinkedIn (Bentzley 2017), and it con- As a concrete example of the need for unbiased cludes by stating, “By. . .driving relentlessly until the forecasts, we summarize the case of a large Cali- forecast has had the bias addressed. . .the organiza- fornian consumer electronics firm, disguised as “Lei- tion can make the most of its efforts and will continue tax,” as reported in Oliva and Watson (2009). This to improve the quality of its forecasts and the supply paper categorizes forecast biases as either intentional

4863 Siegel and Wagner: Profit Estimation Error in the Newsvendor Model 4864 Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS

(driven by misaligned incentives, politics, etc.) or described in section 13.4 of Cachon and Terwiesch unintentional (resulting from procedural and infor- (2012), results in the opportunity to place a second mational blind spots). The authors discuss the imple- order at a higher cost once demand is realized to mentation at Leitax of a coordination system to address satisfy any unmet demand from the first order; we these biases, which results in an increase of forecasting assume that utilizing reactive capacity requires a fixed accuracy, defined as one minus the ratio of the absolute cost of $5,500 and a per-unit premium of 20% so that deviation of sales from forecast to the forecast itself theunitcostofreactivecapacityis(1 + 20%)c 48. from 58% to 88%. These forecasting improvements di- Applying the analysis from section 13.4 of Cachon rectly enhance operational outcomes: average on-hand and Terwiesch (2012)withtheestimated exponential ˆ inventory decreases from $55 million to $23 million, and distribution with mean θ 182.15, we calculate that excess and obsolescence costs decrease from an average we should order 33 units in the original order and of $3 million per year to practically zero. However, this utilize reactive capacity for any demand in excess paper is silent on statistical bias in forecasts (a form of of 33, which results in an expected profit of $4,100. unintentional procedural bias), which is the focus of Therefore, we conclude that reactive capacity is not our research and can supplement the coordination worth it ($4,100 < $4,253). But this would be wrong. system presented in Oliva and Watson (2009). Using the exact exponential distribution with mean We study a scenario in which the demand distri- θ 200, the nonreactive and reactive expected profits bution is specified by a finite number of parameters are $4,670 and $5,041, respectively, and it is beneficial (e.g., mean, standard deviation) that are estimated to utilize the reactive capacity. However, these profit from data. Even though we utilize unbiased and values are typically not available because the true consistent estimators, when fed through the optimi- mean θ is typically not available and is only estimated ˆ zation operator of the newsvendor model, an incor- (with error) as θ 182.15. In our paper, we show that ˆ rect estimation of the optimized expected profitmate- the original profit estimates based on θ 182.15 are rializes. We derive this asymptotic expected estimation biased, and we provide adjustments to correct for the error in closed form for a generic continuous demand biases that can be used by the newsvendor, who does distribution with k parameters that are estimated from not have access to the true mean. Using these ad- ˆ data. We also design an adjustment term, which ac- justments (that only require θ and not the true un- counts for estimation error distortions, that provides known value θ), our bias-adjusted estimates for the an unbiased estimation of the maximized profit(as- nonreactive and reactive expected profits are $3,947 ymptotically to second order) that more accurately and $4,088, respectively, and we make the correct reflects actual realized profit. We also summarize conclusion that reactive capacity is beneficial. Finally, experiments that show that (1) the expected estima- note that the bias adjustment of the (nonreactive) tion error is statistically significant, and (2) our ad- newsvendor profitis7.2%ofitsbiasedestimateand justed profit has no statistically significant expected thatthetrueexpectedprofit(withrespecttoθ)of estimation error. Finally, we also consider expected using reactive capacity is 8.0% higher than the profitof estimation error in the order quantity and derive its not using reactive capacity. general asymptotic functional form; in particular, we Our contributions for identifying the newsvendor show that (1) when demand is exponentially or nor- profit’s expected estimation error are derived within mally distributed, the order quantity has no expected the theoretical framework developed by Siegel and estimation error, and (2) when demand is log-normally Woodgate (2007) for quantifying the expected esti- distributed, the order quantity has a expected estimation mation error in the performance of financial portfolios error that can be corrected. optimized using estimates of true parameters; the We provide the following numerical example, moti- newsvendor model shares the basic structure of op- vated by valuing reactive capacity in a newsvendor con- timizing while pretending that estimates are correct. text, which illustrates the practical benefits of our research. Achieving our closed-form profit adjustment is com- Example 1. Consider a newsvendor situation in which plex because the order quantity and the exact func- the unit selling price p 100, the unit procurement cost tional form of the realized profitarebothnonlinear c 40, and demand is exponentially distributed with a functions of the estimation error of the parameter(s). mean θ 200. Suppose that we have the following 10 We use the method of statistical differentials to find sample demands from this distribution {217, 444, 148, the second order Taylor series approximation to the 219, 251, 126, 28, 32, 210, 147}, which lead to an esti- expected future profit (averaged over the random ˆ mated mean θ 182.15. Using an estimated exponential future demand realization) and then take its expec- ˆ distribution with mean θ 182.15, the optimal news- tation (over the sampling error of parameter estima- vendor order quantity is 167, which results in an es- tion). Our result is asymptotically correct when the timated expected profit of $4,253. Reactive capacity, as sample size used for demand-function estimation is Siegel and Wagner: Profit Estimation Error in the Newsvendor Model Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS 4865 large (and simulations demonstrate that our results 1.2. Literature Review are useful even for moderate sample sizes) and re- The newsvendor is a fundamental model of opera- mains statistically consistent when estimated values tions management, appearing frequently in both prac- are substituted for the unknown parameter(s) of the tice and academic research. Consequently, there is a demand distribution. In effect, we use perturbation vast literature on various aspects of the newsvendor; analysis to discover how estimation errors are misused here, we discuss the most relevant papers to properly by the newsvendor’s optimization process, which position our contributions. To the best of our knowl- wrongly “believes” that the estimated parameters are edge, we are the first to document expected estimation correct. All proofs are provided in the online elec- error in the maximized expected profit perceived by tronic companion. the newsvendor, who estimates, using available data, the parameters of a demand distribution. Much at- 1.1. Contributions tention has been focused on various consequences of Our research provides the following main contributions: distributional uncertainty; in particular, we discuss • For a generic demand distribution, under ap- the following streams of literature: (1) the data-driven propriate assumptions, we derive a second order newsvendor, (2) approximation algorithms for sto- approximation of the newsvendor profit’sexpected chastic inventory-management problems, (3) algorith- estimation error for an arbitrary order quantity that mic inventory management, (4) distributionally robust is a smooth function of the estimated parameters. inventory management, and (5) the newsvendor with Using these expected estimation error results, we operational statistics in which estimation and optimi- provide an adjusted expected profit expression that zation are performed jointly. However, as we discuss in can be computed without knowledge of the true the following, these streams focus on identifying good parameter(s), which is an asymptotically unbiased ordering strategies but have not studied the expected fi estimator of the true expected profitwhentheesti- pro t estimate that a decision maker would also require mated order quantity is used. Hypothesis testing in practice. experiments confirm our theoretical results in that we show at the 5% test level the unadjusted estimation 1.2.1. Data-Driven Newsvendor. The newsvendor model error is significantly positive (the null is rejected), can be solved using only data without any assumptions whereas the adjusted estimation error is not signifi- on the form of the demand distribution: Kleywegt cantly different from zero (the null is accepted). et al. (2002) design the sample average approxima- • We similarly derive expected estimation error tion (SAA) method using available data to create an expressions for the order quantity, which allows us to empirical demand distribution, and the optimal order also provide adjusted order quantities. We show that quantity is found at the critical fractile of this dis- some distributions (e.g., exponential and normal) tribution (in the newsvendor context); these authors already exhibit order quantities with zero expected also identify a bias in a natural estimation of the estimation error, but other distributions (e.g., log- optimal objective value (for more general stochastic normal) result in a nonzero expected estimation er- discrete optimization problems) but do not provide ror. We provide an adjusted order quantity that is an an adjustment term to correct the bias as we do in our asymptotically unbiased estimate of the true optimal paper. Levi et al. (2007a) study the sampling-based order quantity. Again, hypothesis testing experiments SAA method for the newsvendor as well as a mul- confirm our theoretical results. tiperiod extension of it and show that their policy • We characterize the relative importance of profit has a solution that is provably near optimal with high and order quantity expected estimation errors. In probability; see the references therein for a more particular, the adjustment to the order quantity re- comprehensive literature review related to the data- sults in a negligible change in expected profit: for a driven newsvendor. Levi et al. (2015)refine the anal- log-normal distribution of demand, the adjustment ysis in Levi et al. (2007a) to produce tighter perfor- results in a 0.001% increase in profitinanexample.In mance bounds by introducing an additive bias into contrast, correcting the profit expected estimation the order quantity; in addition, they utilize a second error is not negligible: for the same log-normal dis- order Taylor series in their analysis as we do in ours. tribution of demand, the adjustment results in a 1.1% Ban and Rudin (2019) extend the data-driven ap- increase in profit. Furthermore, when demand is proach of Levi et al. (2007a, 2015)toincludefea- exponentially distributed, the order quantity is un- tures (explanatory variables) that influence the de- biased, but the profit estimation error in an example is mand distribution, using techniques from machine 3.4% of the true expected profit. Thus, the profit learning; as in Levi et al. (2007a), performance bounds adjustments can be managerially relevant. are derived that hold with a prescribed probability. Siegel and Wagner: Profit Estimation Error in the Newsvendor Model 4866 Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS

He et al. (2012) consider a similar features-based These papers focus on different aspects of a single newsvendor model for staffing hospital operating location dealing with a single product; Levi et al. rooms. However, in these references, thebenchmark (2017) develop various approximation algorithms with optimal expected profit depends on the true distri- performance guarantees between two and three for bution, which is unknown, and hence, the optimal multiechelon systems with multiple products. How- expected profit is not calculable. The only calculable ever, all these approximation algorithm results re- profit expression is with respect to the data-driven quirecompleteknowledgeofthetruedemanddis- empirical distribution, which is the realization of a tribution to calculate the expected profits—an assumption random variable (because of sampling), and the main we relax. results also calculate the expected value of this ran- dom variable with respect to the true unknown dis- 1.2.3. Algorithmic Inventory Management. There are a tribution. Thus, although the performance guarantees number of other papers that take an algorithmic ap- of these papers are powerful, they are implicit results, proach to solving data-driven inventory-management and no calculable expressions exist for the expected problems. Kunnumkal and Topaloglu (2008) apply profits. In contrast, we focus on natural calculable a stochastic gradient approach to a multiperiod news- profit expressions based on distributional parameters vendor model. Burnetas and Smith (2000)andHuh estimated from data, which turn out to have nonzero and Rusmevichientong (2009) also apply stochastic expected estimation errors, and we provide adjust- gradient algorithms except, to a scenario with cen- ments to the biased expressions, thereby giving the sored demand in which only sales data—not true decision maker a calculable unbiased estimate of demand data—are available; Godfrey and Powell expected profit.BanandRudin(2019) also interpret (2001), Huh et al. (2011), and Besbes and Muharre- their results in terms of “generalization error,” in moglu (2013)alsostudytheimpactofcensoredde- which the decisions are based on in-sample data, but mand for the newsvendor via various algorithms. the quality of the decisions is evaluated out-of-sample Hannah et al. (2010) study a more general stochastic (in terms of the unknown true distribution), and their optimization problem algorithmically, which can be analysis utilizes results from machine learning on applied to an inventory-management context. Our generalization error (see references therein); in this work differs from these papers in that we focus on a interpretation, we provide explicit closed-form cor- simpler, single-period model so that we may obtain rections for evaluating the out-of-sample expected closed-form solutions, and we assume that demand, profits, a result that is new to the literature to the best not just sales, data are available. of our knowledge. Ban et al. (2019) also study features of demand, but the focus is on algorithmic solutions 1.2.4. Distributionally Robust Inventory Management. The to multiperiod inventory-management models, whereas sensitivity of the newsvendor solution to the demand we focus on a single-period model in which closed-form distribution has been studied in the distributionally solutions for the order quantity, profitexpressions,and robust optimization literature. Scarf (1958)analyzes corrections are available. the so-called “distribution-free” newsvendor model, in which the mean and standard deviation of de- 1.2.2. Approximation Algorithms for Stochastic Inven- mand are given, but the distribution is not; the ob- tory Management Problems. A related stream focuses jective is then to maximize the worst-case expected on approximation algorithms for various stochastic profit, which is minimized over all distributions with inventory management models with provable per- the given mean and standard deviation. Gallego formance guarantees that do not require a probabi- and Moon (1993) provide extensions and a review listic qualifier (i.e., the performance guarantees hold of similar work. More recently, Perakis and Roels with probability one). Levi et al. (2007b)providetwo- (2008)andNatarajanetal.(2018) consider similar and three-approximation algorithms for the multi- but more sophisticated distribution-free settings in period periodic-review stochastic inventory-control which, again, the focus is to derive order quantities and the stochastic lot-sizing problems, respectively. that serve well in the worst case. These papers as- Levi et al. (2008b) design a two-approximation al- sume that their information is correct and do not gorithm for a multiperiod capacitated inventory control consider the impact of estimation error, which we model. Levi et al. (2008a) derive two-approximation identify as an assumption with consequences to the algorithms for stochastic inventory control models newsvendor’s anticipated expected profit. with lost sales. Levi and Shi (2013) provide a three- approximation algorithm for a multiperiod stochastic 1.2.5. Operational Statistics. In the paper perhaps most lot-sizing problem with order lead times, and Shi related to our research, Liyanage and Shanthikumar et al. (2014) design a four-approximation algorithm (2005) consider the single-period newsvendor model for a similar problem with capacities and setup costs. under a parametric demand distribution in which Siegel and Wagner: Profit Estimation Error in the Newsvendor Model Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS 4867 data are available to estimate parameters. Assuming profit for an arbitrary ordering quantity q,whenthe that demand is exponentially distributed, the authors truth θ is known, is as follows: intentionally bias the order quantity to obtain higher () []() π ,θ , − expected profit via a joint estimation–optimization q pE∫X∼F()·,θ min q X cq q []() operation. We also consider the special case in which p xf() x,θ dx + pq 1 − Fq,θ − cq (2) demand is exponentially distributed, and although 0 ∫ the operational-statistics order quantity improves () q − − (),θ , profit, we show that it still exhibits a nonzero ex- p c q p Fx dx 0 pected estimation error. Furthermore, we show that the expected estimation error is considerably larger, where the last equality is by integration of parts. asymptotically, than the profit improvement. Chu Note that all of the results in our paper continue to et al. (2008) study the operational-statistics approach hold for the case of a fixed salvage value s. Instead using Bayesian analysis for parametric distributions of p min(q, X)−cq,theprofit becomes p min(q, X)− characterized by location and scale parameters, deriving cq + s[q − min(q, X)]. Rearranging, the profitis(p − s) closed-form solutions for an exponential distribution of min(q, X)−(c − s)q, and so the newsvendor problem demand (matching that of Liyanage and Shanthikumar with salvage value is entirely equivalent to the news- 2005) as well as for a uniform distribution of demand. vendor problem without salvage value provided that the In contrast, our results are not limited to location- cost and price are both reduced by the salvage value. For scale families of distributions and are applicable to a ease of exposition, we assume s 0 in our paper. broad array of distributional families for which we derive closed-form asymptotic expressions for the 2.1. Motivation for a Parametric Model fi ’ pro t s expected estimation error, which can be used There are a number of reasons a manager should to easily correct the error. consider using a parametric model over the purely data-driven approaches reviewed in Section 1.2.1 2. The Newsvendor Model (e.g., the SAA method). Researchers at Amazon, We let X denote random demand, which we assume from their Forecasting Data Science and Supply Chain has a continuous distribution with cumulative dis- Optimization Technologies groups, discuss how para- tribution function F(x,θ),whereθ is a parameter that metric distributions fit within their broader forecast- specifies the demand distribution from a family of ing activities (Madeka et al. 2018). In particular, distributions. For this introduction, we assume that θ they discuss, in a newsvendor context, how (shifted) is a scalar though we extend all our results to the case gamma and log-normal distributions may be utilized in which θ is a finite-dimensional vector. The prob- to forecast future demand. In particular, they point ability density function of demand for a given θ is out that “. . .the lognormal distribution is suitable for ( ,θ) ∂ ( ,θ) evergreen products with non-negligible demands.” f x ∂x F x andisassumedtoexistandbe positive on a contiguous interval of support. The Furthermore, the expected profitislarger(closerto economics of the newsvendor model are as follows: optimal) with a parametric model when such a model p > 0isthesalespriceperunit,c > 0isthecostper can reasonably be assumed (e.g., by examining his- unit, and q ≥ 0 is the order quantity; we assume p > c tograms of data and/or more rigorous distributional throughout. The newsvendor profit is the random goodness-of-fit hypothesis testing). An efficient es- variable p min(q, X)−cq, and we focus on the ex- timator of the order quantity, for instance, using the pected profit maximum likelihood estimator (MLE) for a para- () []() metric model, has minimum asymptotic variance π ,θ , − . among all estimators (see sections 6.2–6.4 of Lehman q pEX∼F()·,θ min q X cq 1983 for details and regularity conditions). The de- crease from optimal expected profit is proportional This expected profit is known to be maximized when q (asymptotically) to the variance of this estimator as is chosen so that 1 − F(q,θ)c, where the left-hand p may be seen using the expectation of a second order side of this equation is the probability of selling the qth Taylor series expansion of profit as a function of the unit; we may write this optimality condition as () estimated order quantity. For example, measuring fi − c asymptotic statistical ef ciency as the ratio of vari- q()θ []F()·,θ 1 1 − , (1) fi p ances, if the SAA estimate has ef ciency 65% relative to a (correct) parametric model, then the drop in where the inverse function is taken with respect to the expected profit(ascomparedwithusingtheoptimal firstargumentofF at a given value of θ. The expected but unknown order quantity) using the MLE is only Siegel and Wagner: Profit Estimation Error in the Newsvendor Model 4868 Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS

65% as large as the drop in expected profitincurred an asymptotically unbiased estimate of E[π(q˜,θ)] that when using the SAA estimator. In fact, under expo- does not require knowledge of the true value of θ, nentially distributed demand, the asymptotic statis- where q˜ is an arbitrary order quantity defined as a ˆ tical efficiency of the SAA order quantity with respect smooth function of θ. We are abusing notation here ˜ ˜ to the parametric order quantity is never larger than for simplicity of explanation: in fact, q qn,ase- 65%; we show this formally in the next section. quence of functions that we assume converges to the Finally, in a biological context, Jabot (2015)argues constant q(θ) in the limit for a large sample size n for ˆ that parametric forecasting is preferred to nonpara- which θ is converging to θ. metric forecasts because of (1) the ability to diagnose In addition, our results have implications for im- parametric forecasting failure via Bayesian model proving the estimation of the expected mismatch checking procedures, and (2) forecasting uncertainty cost, which is defined as cE[max{q − X, 0}] + (p − c) can be estimated using synthetic data generated from E[max{X − q, 0}]. In particular, it is straightforward to the fitted parametric model. show that the expected mismatch cost is equivalent to (p − c)E[X]−π(q,θ); see, for instance, section 13.1 of 3. The Impact of Parameter Estimation on Cachon and Terwiesch (2012). Because expected de- Expected Profit mand E[X] is estimated unbiasedly (assuming the θ In practice, the parameter θ can be unknown, in which mean is included in the parameter vector ), any ˆ fi fl case it is estimated as θ (i.e., using data). The news- bias in forecasting expected pro t ows directly and ˆ vendor, using the estimated distribution F(·, θ), chooses dollar-for-dollar as a bias of the expected mismatch ˆ the stock quantity qˆ ≜ q(θ).Theparameterestimateis cost. It, therefore, follows that, for the naive news- ˆ assumed to be unbiased and consistent so that E[θ] vendor, the expected mismatch cost is estimated with θ [(θˆ − θ)2] (1) bias and that, by using our profitbiasadjustment,one and E O n ,wheren is the size of an independent and identically distributed sample of can obtain an (asymptotically) unbiased estimate of the expected mismatch cost. demand quantities Xi ∼ F(·,θ), i 1, ..., n. Detailed assumptionsaregiveninSection3.2. Finally, we apply our Taylor series technique to Actual expected profitisthenπ(qˆ,θ),whichis derive asymptotically unbiased estimates of the true optimal order quantity that also depend on the un- computed using the true parameter value that de- θ termines demand; that is, the order quantity is de- known value of to achieve the following result: ˆ [] termined using the estimated demand parameter θ, E qˆ − Adjustment q()θ + o()1/n . but profitisaveragedoveractualdemandX with the (·,θ) correct distribution F . However, the decision In the next two Sections 3.1 and 3.2, we focus on the θ maker does not have access to , and therefore, the expected profit adjustment and consider the cases of decision maker does not have access to the expected an exponential distribution of demand and a general fi π(ˆ,θ) pro t q . The only calculation a decision maker distribution of demand, respectively. In the expo- can reasonably do, initially, to have an idea what nentialcase,weareabletoderiveanexactadjustment fi π(ˆ, θˆ ) expected pro t will actually be is q ,whichis (i.e., no o(1/n) error term). In the general case, under (·, θˆ ) calculated using the estimated distribution F ;we appropriate assumptions, we derive an asymptotic π(ˆ, θˆ ) fi call q the perceived expected pro t. Note that adjustment via Taylor series analysis. both of these expected profits (actual and perceived) θˆ are random variables, depending on the estimate 3.1. Closed-Form Solutions for an Exponential and its random estimation error, generated when the Distribution of Demand truth is θ. ˆ ˆ In this section, we consider the case in which demand In this paper, we show that π(q, θ) is a biased es- θ fi π(ˆ,θ) is exponentially distributed∑ with unknown mean timator of the true expected pro t q .Amain θˆ 1 n that is estimated as n i1 Xi. In this case, the order contribution of our paper is to derive an adjustment to ˆ ( / )θˆ ˆ ˆ ˆ quantity is q ln p c .Theorderquantityq is of the π(q, θ) that results in an asymptotically unbiased ˜(θˆ ) θˆ fi form q a ,wherea is a constant; many of the estimate of the true expected pro t despite the in- following results are presented in terms of the more θ ˆ ˆ accessibility of itself: general q˜(θ)aθ. In our literature review, we de- []() []() ˆ – E π qˆ, θ − Adjustment E π qˆ,θ + o()1/n , scribe the joint estimation optimization procedure studied in Liyanage and Shanthikumar (2005), which ≜ [( / )1/(n+1) − ]θˆ where the expectation is taken with respect to the derives qOS n p c 1 as the optimal order sampling distribution. This adjustment is calculated quantity for an exponential distribution of demand via Taylor series expansions and can be used without (the subscript OS standing for operational statistics). ˆ ˆ knowledge of the true value of θ. In addition, our Thus, the general form of ordering quantity, q˜(θ)aθ, analysis is not limited to the order quantity qˆ;weprovide unifies the classical and the operational statistics Siegel and Wagner: Profit Estimation Error in the Newsvendor Model Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS 4869

ˆ ˆ fi methods because q aCθ with aC ln(p/c) and qOS perceived expected pro t has smaller variance than the ˆ 1/(n+1) fi aOSθ,whereaOS n[(p/c) − 1].Ourfirst result, unadjusted perceived expected pro t when the classical from Liyanage and Shanthikumar (2005), provides order quantity (a aC) is used. the actual expected profit with respect to the true θ mean averaged over the sampling distribution for 3.1.1. Asymptotic Properties for the Exponential ˜(θˆ ) θˆ the general order quantity q a ;weprovideanew Distribution. We next develop the asymptotic proper- proof of this result. Note that we are unaware of any ties of the adjustment’s improvement and of the ex- similar result in the literature for a distribution other pected profit improvement from using the OS or- than the exponential distribution. ˆ 1/(n+1) ˆ der quantity qOS aOSθ n[(p/c) − 1]θ instead of ˆ ˆ ˆ Lemma 1 (Liyanage and Shanthikumar 2005). The overall the classical choice q aCθ ln(p/c)θ.Thisasymp- expectation of actual expected profit for exponentially dis- totic analysis allows us to show that, although the OS ˆ ˆ tributed demand with order quantity q˜(θ)aθ and con- expected profit improvement becomes negligible at stant a at a fixed sample size n is moderate sample sizes, the adjustment’s improve- []()() []() ment remains economically meaningful in this range. ˆ n n E π q˜ θ ,θ p − ac − p θ. n + a Proposition 2. For exponentially distributed demand, the asymptotic expected profit improvement from using the OS Our next result provides the perceived expected profit ˜ ˆ ˆ order quantity in place of the classical choice is for the general order quantity q(θ)aθ,wherethe []()[]() (·, θˆ ) ˆ estimated distribution F is used instead of the E π qOS,θ − E π q,θ true unknown distribution F(·,θ). ()()[]() θ − / + / 2 2 () c 2ln p c ln p c 1 fi + O . Lemma 2. The overall expectation of perceived expected pro t 8n2 n3 for exponentially distributed demand with order quantity ˆ ˆ q˜(θ)aθ and constant a at a fixed sample size n is From Proposition 2, we see that the actual expected []()() [] profit improvement from using q instead of qˆ is π ˜ θˆ , θˆ − − −a θ. OS E q p ac pe O(1/n2). The next two results provide asymptotic

expressions for the profit expected estimation er- Comparing the overall expectation of actual and rors when the unadjusted perceived expected profits perceived expected profitexpressionsofLemmas1 are used to estimate the actual expected profits and 2, we observe that they are clearly different. Their for both qˆ and q ordering quantities, respectively. difference suggests an adjustment for the perceived OS These errors can also be interpreted as improvements expected profit when the true mean θ is not known, when the adjusted expected profits are used instead, and we propose an adjusted expected profit which we contrast with the profit improvement of ()() ()() []() ˆ ˆ ˆ ˆ ˆ n n − Proposition 2. πˆ q˜ θ , θ ≜ π q˜ θ , θ − pθ − e a . (3) a n + a Proposition 3. For exponentially distributed demand, the Note that this expression does not require knowledge asymptotic expected estimation error for the unadjusted ofthetruevalueθ (which contrasts with Lemmas 1 perceived expected profit when the classical order quantity is and 2) and only requires its unbiased and consistent used is ˆ []() () estimator θ.Thisadjustedexpectedprofitisanun- ˆ E π qˆ, θ − π qˆ,θ biased estimator for the actual expected profitas [][]() []() []() () 2 3 4 presented in the next proposition. In addition, with ln p/c −8lnp/c + 3lnp/c 1 cθ + + O , the classical order quantity, this adjusted expected 2n 24n2 n3 profit has lower variance than the unadjusted ex- pected profit, thereby reducing the mean squared and when the OS order quantity is used, it is error (the sum of variance and squared bias) by re- []()() ˆ ducing both contributions. E π q , θ − π q ,θ OS OS []() []() ⎡⎢ 2 3⎤⎥ Proposition 1. ⎢ −24 ln p/c + 16 ln p/c ⎥ If demand is exponentially distributed with ⎢[]() []() ⎥ fi ˜(θˆ ) θˆ ⎢ / 2 − / 4 ⎥ xed sample size n and order quantity q a , then ⎢ ln p c 3lnp c ⎥ cθ⎢ + ⎥ the adjusted perceived expected profit, defined in Equation (3), ⎢ 2 ⎥ ⎢ 2n 24n ⎥ is an unbiased estimator for the actual expected profit, ⎣⎢ ⎦⎥ ˆ ˜ ˆ ˆ ˜ ˆ E[πa(q(θ),θ)] E[π(q(θ),θ)], and the adjustment is neg- () fi 1 ative, decreasing the perceived expected pro t and thereby + O . correcting for its over-optimism. In addition, the adjusted n3 Siegel and Wagner: Profit Estimation Error in the Newsvendor Model 4870 Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS

From Proposition 3, we observe that the asymptotic 3.1.3. Efficiency of Parametric and SAA Estimates of expected estimation error for the OS order quantity is Order Quantity for the Exponential Distribution. In this identical to that of the classical order quantity, up to section, we formalize our argument from Section 2.1 O(1/n), and differs only in the O(1/n2) term. Fur- that the parametric estimate of the optimal order thermore, these results show that the expected profit quantity is asymptotically more efficient than the ˆ − / estimation errors for q and qOS are both O(1/n),which SAA estimate, which orders the 1 c p percentile of is asymptotically greater than the actual profitim- the empirical distribution. The statistical efficiency of 2 ˆ ˆ fi provement of O(1/n ).Thus,ifoneweretouseqOS the estimator qSAA with respect to the MLE q is de ned Var(qˆ) instead of qˆ,asmallamountofadditionalprofit as (ˆ ). Var qSAA would be realized; however, if a naive estimation of ˆ Lemma 3. The asymptotic statistical efficiency of the SAA expected profit is used (i.e., using F(·, θ)), the expected order quantity qˆ as compared with the MLE order quantity estimation error dwarfs the improvement. Therefore, ˆ SAA qˆ θ ln(p/c) under the exponential demand parametric we recommend decision makers adopt our unbiased fi model tends asymptotically (as sample size n grows) to adjusted expected pro t expression to eliminate the () []() expected estimation error whether or not they use the ˆ / 2 Var()q → ln p c , OS order quantity. ˆ / − Var qSAA p c 1

3.1.2. Numerical Verification for the Exponential which can be no larger than 64.8%. Distribution. In this section, we (numerically) com- pare the magnitude of the operational statistics 3.2. Asymptotic Solutions for a General approach’sprofit improvement over the classic order Distribution of Demand quantity (solid line) with that of its expected esti- In this section, we extend our results from the ex- mation error (dashed line) for demand that is expo- ponential distribution to a larger class of smooth nentially distributed with mean θ 200. In the left distributions with smoothness conditions given as pane of Figure 1,forc 0.4andp 1, we see the follows. Furthermore, we consider an arbitrary or- ˆ ˆ improvement and expected estimation error as a der quantity q˜(θ),whereq˜ isasmoothfunctionofθ in ˜ θˆ function of sample size n,andweobservethat,al- the sense that q is differentiable as a function of ,and ˆ though the profit improvement of the operational the derivative is locally Lipschitz continuous in θ.We ˆ statistics approach is indeed positive, the profitex- also assume that q˜(θ) is a consistent estimator of q˜(θ) ˆ pected estimation error still exists and is much larger in the sense that q˜(θ) converges in probability to q˜(θ) than the improvement. Furthermore, the profitim- as sample size n grows—an example of which appears provement becomes negligible for moderate values as qOS in Proposition 2. For such a distribution and of n, yet the expected estimation error remains eco- order quantity, using a Taylor series approximation, we ˆ ˆ ˆ nomically meaningful for much larger values of n.In expand the expression for π(q˜(θ), θ)−π(q˜(θ),θ) about ˆ therightpane,weprovidetheprofit improvement θ to second order in θ − θ and then take the expected and expected estimation error as a function of c/p for value; the result is the second order expected estimation n 20 and we observe similar results. error,whichisO(1/n).Welet∇ denote the gradient

Figure 1. (Color online) Demand Has an Exponential Distribution with θ 200

Notes. (Left) Operational statistics profit improvement (over classic order quantity) and profit expected estimation error as a function of n. (Right) Operational statistics profit improvement (over classic order quantity) and profit expected estimation error as a function of c/p. Siegel and Wagner: Profit Estimation Error in the Newsvendor Model Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS 4871

operator with respect to θ and let ∇2 denote the k-dimensional parameter θ (θ , ... ,θ ) with unbiased ˆ 1 k Hessianmatrixoperatorwithrespecttoθ. estimator θ equals [ ( ∫ )] Our smoothness conditions (which, henceforth, are ()() q˜()θ ˆ 1 − p · tr Cov θ ∇q˜()θ ()∇F + ∇2Fx(),θ dx to be assumed) on the parametric family of distri- 2 butions are adapted from Lehman (1983,chapter6, () 0 fi 1 corollary 2.3, theorems 2.3 and 1.1) and are suf cient + o , to ensure that the maximum likelihood estimator n exists and is efficient. In particular, these assumptions where tr denotes the matrix trace operator, ∇q˜(θ) is the are satisfied by any one-parameter exponential family gradient of q˜(θ) with respect to θ, ∇FisthegradientofF of distributions (see, e.g., Lehman 1983). with respect to θ and evaluated at (q˜(θ),θ), ∇2F(x,θ) is the ˆ k × k Hessian matrix of F with respect to θ,andCov(θ) is ˆ 3.2.1. Assumptions for General Demand Distribu- the covariance matrix of θ. tion Family. The expression identified in Proposition 4 provides 1. The parameter space for θ is an open interval evidence that a nonzero expected estimation error (possibly unbounded), and the distribution F(·,θ) has indeed exists; we provide further evidence via nu- density f (·,θ). The observations are independent and merical experiments, focused on identifying statisti- identically distributed with distribution F(·,θ) for cally significant nonzero expected estimation errors, some true value of θ that is in the parameter space. in Section 3.5. 2. The distributions F(·,θ) have common support We can apply Proposition 4 to the special case in A {x : f (x,θ) > 0} and are distinct to ensure esti- ˆ which q˜(θ)qˆ for which ∇qˆ(θ)−∇F/f from Lemma mation is possible. EC.4, which results in the following corollary. 3. For every x ∈ A, the density f (·,θ) is three times ˆ differentiable with respect to θ,andthethirdderiv- Corollary 1. The profit expected estimation error E[π(qˆ,θ)− ative is continuous∫ in θ. π(qˆ,θ)] for qˆ for the case of a k-dimensional parameter θ (θ , ...,θ )’ θˆ 4. The integral f (x,θ)dx can be twice differenti- 1 k with unbiased estimator equals ated under the integral sign. []()∫ () ()()∇ ()∇ q 5. The Fisher information I(θ) is positive and finite · θˆ F F − 1 ∇2 (),θ + 1 ,

p tr Cov Fx dx o (see Lehman 1983, section 6.4 for additional assump- f 2 0 n tions for the vector parameter case). where q q(θ), tr denotes the matrix trace operator, 6. For any given θ in the parameter space, there 0 f f (q,θ), ∇F is the gradient of F with respect to θ and exists ν>0andafunctionM(x) (both of which may evaluated at (q,θ), ∇2F(x,θ) is the k × k Hessian matrix of θ θ [ ( )] < ∞ |∂3 depend on 0)suchthatE 0 M X and also θ (θˆ ) 3 F with respect to ,andCov is the covariance matrix ln f (x,θ)/∂θ |≤M(x) for all x ∈ A and θ0 −ν<θ<θ0 +ν. ˆ of θ. 7. The probability∑ of multiple roots to the likeli- hood equation ∂ ln[f (x ,θ)]/∂θ 0 tends to zero as The next corollary considers the scalar θ case of i ˜ ˆ thesamplesizen →∞. Proposition 4 for the general ordering quantity q(θ). ˆ In the following proposition, for a multidimensional Corollary 2. If q˜(θ) is smooth as defined at the start of the θ (θ , ...,θ ) fi ˆ parameter 1 k ,wherek is a nite integer, section, then the profit expected estimation error E[π(q˜(θ), ’ ˆ ˆ ˆ we show that the newsvendor s perceived expected θ)−π(q˜(θ),θ)] for q˜(θ) for the case of a one-dimensional fi π(˜(θˆ ), θˆ ) ˆ pro t q is systematically biased in that it parameter θ with unbiased estimator θ equals fi differs, on average, from the actual expected pro t ()∫ () π(˜(θˆ ),θ) () q˜()θ q that is realized based on the true, but un- − · θˆ · ˜ ()θ + 1 (),θ + 1 , θ p Var q Fθ Fθθ x dx o known, demand parameter . Subsequently, we show 2 0 n how these expected estimation errors can be used to ∂q˜(θ) ∂ ∂2F(x,θ) create adjusted expected profits that more accurately where q˜ (θ) ,Fθ F (q˜(θ),θ),Fθθ(x,θ) , ˆ ∂θ ∂θ ˆ ∂θ2 reflect reality (i.e., they are unbiased). In addition, and Var(θ) is the variance of θ under its sampling note that the sign of the expected estimation error is distribution. not yet evident; we explore the direction of the error in Note that Corollary 2 is applicable to both the Section 3.3 by applying our generic expressions to ˆ − naive order quantity qˆ [F(·, θ)] 1(1 − c/p) as well specific distributions. as the operational statistics order quantity qOS n ˆ /( + ) ˆ Proposition 4. If q˜(θ) is smooth as defined at the start [(p/c)1 n 1 − 1]θ for an exponential distribution of of the section, then the profit expected estimation er- demand. Furthermore, the following corollary results ˆ ˆ ˆ ˆ ˆ ror E[π(q˜(θ), θ)−π(q˜(θ),θ)] for q˜(θ) for the case of a when assigning q˜(θ)qˆ because q˜ (θ)−Fθ/f for the Siegel and Wagner: Profit Estimation Error in the Newsvendor Model 4872 Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS ˆ ˆ ˆ naive order quantity as follows by differentiating E[π(q˜(θ), θ)−π(q˜(θ),θ)] for q˜(θ) for the case of a ˆ θ (θ , ... ,θ ) the equation 1 − c/p F(q(θ),θ) with respect to θ to k-dimensional parameter 1 k with unbiased ˆ find 0 f (qˆ(θ),θ)qˆ (θ)+Fθ(qˆ(θ),θ)f qˆ (θ)+Fθ. estimator θ equals Corollary 3. The profit expected estimation error E[π [ ˆ ()[]() (ˆ, θ)−π(ˆ,θ)] ˆ p 2 −1 q q for q for the case of a one-dimensional tr E ∼ ()·,θ ∇ ln fX(),θ θ θˆ n X F parameter with unbiased estimator equals ()∫ ] () () q˜()θ ()∫ () 1 2 1 () 2 q ×∇q˜()θ ()∇F + ∇ Fx(),θ dx + o , ˆ Fθ 1 1 p · Var θ · − Fθθ()x,θ dx + o , 2 0 n f 2 0 n

2 and the case of a one-dimensional parameter θ with un- (θ) ∂F(q,θ) ( ,θ) ( ,θ)∂ F(x,θ) where q q ,Fθ ∂θ ,f f q ,Fθθ x ∂θ2 , θˆ ˆ ˆ biased estimator equals and Var(θ) is the variance of θ under its sampling distribution. []p ∂2 (),θ nEX∼F()·,θ ∂θ2 ln fX 3.2.2. Estimating the Covariance Matrix. Our framework ()∫ () ( , ..., ) q˜()θ assumes that there is a single vector of data, X1 Xn , × ˜ ()θ + 1 (),θ + 1 . θ (θ , ...,θ ) q Fθ Fθθ x dx o that is available to approximate 1 k ,which 2 0 n θˆ (θˆ , ..., θˆ ) leads to a single vector estimate 1 k . However, this setup is not sufficient to directly esti- Corollary 4 allows us to derive adjustments for the ˆ mate the covariance matrix Cov(θ) required for Prop- newsvendor’s perceived expected profit that are easily osition 4 and Corollary 1 becausewewouldneed computable to better reflect reality; we explore this ˆ multiple samples of the vector estimate θ.Conse- topic in the next section. ˆ quently, we approximate Cov(θ) by using the Fisher information matrix 3.2.3. Unbiased Adjusted Expected Profit Estimation. In []() fi ()θ − ∇2 (),θ , this section, we provide an adjusted expected pro t I EX∼F()·,θ ln fX πˆ fl estimator, denoted as a,whichbetterre ects the actual expected profit observed by the newsvendor where the expectation is taken over the random de- than its naive estimate, in which the hat over the π mand X with distribution F(·,θ), parameterized by indicates that this is an estimated value that is available to thetruevalueθ,and∇2 is the Hessian second de- the newsvendor. To construct this adjusted expected rivative operator with respect to θ. Using the as- profit, we modify the newsvendor’s perceived expected ymptotic theory of maximum likelihood estimation ˆ ˆ profit π(q˜(θ), θ) by subtracting a consistent estimate of (see, for instance, Lehman 1983,sections6.2–6.4, in the second-order expected estimation error that is particular theorems 2.3 and 4.1) together with our obtained by replacing unknown quantities in the assumptions for general demand distribution family error formulas with their estimates and ignoring the near the start of Section 3.2, we may approximate the ˆ o(1/n) error term. The result is that this adjusted ex- covariance matrix of θ as follows: pected profit is unbiased (to the second order) with re- () ˆ () fi π(q˜(θ),θ) ˆ 1 − 1 spect to the actual expected pro t in the sense Cov θ I 1()θ + o . n n that their expectations are equal to the second order. For the case of a one-dimensional parameter θ with ˆ Similarly, for the univariate parameter case, the Fisher unbiased estimator θ, we propose the following ad- information is the scalar justed expected profit [] ∂2 ()() ()() ()θ − (),θ , ˆ ˜ ˆ ˆ ˜ ˆ ˆ []p I EX∼F()·,θ ln fX πa q θ , θ ≜ π q θ , θ − () ∂θ2 ∂2 , θˆ nE ∼ ()· ,θˆ ∂θ2 ln fX ()X F ˆ ∫ ˆ θ () q˜()θ () which allows the variance of to be approximated as 1 ˆ × ˜ θˆ ˆ + , θˆ , Var(θ) 1 + o(1). The following corollary uses these q Fθ Fθθ x dx nI(θ) n 2 0 expressions to rewrite the expected estimation errors. ˜(θˆ ) fi ˆ ∂F (˜(θˆ ),θˆ ) ( , θˆ )∂2F ( , θˆ ) Corollary 4. If q is smooth as de ned at the start of where Fθ ∂θ q and Fθθ x ∂θ2 x are both section 3.2, then the profit expected estimation error available without knowledge of the true value of θ. Siegel and Wagner: Profit Estimation Error in the Newsvendor Model Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS 4873

Similarly, for the case of a k-dimensional vector pa- exponential distribution, with that of Proposition 3, ˆ rameter θ with unbiased estimator θ,wepropose which is derived using the form of the exponential ()() ()() distribution. Note that the 1 term is identical in both πˆ ˜ θˆ , θˆ ≜ π ˜ θˆ , θˆ n a q q expressions, and the value of using Proposition 3 is [() []()()−1 fi / 2 − p ∇2 , θˆ that the coef cient of the 1 n term is known exactly, tr E ∼ ()·,θˆ ln fX n X F whereas it is not known exactly in Proposition 5, ()]∫ ˜ θˆ using the more general result. ()() q() () ˜ ˆ ˆ 1 2 ˆ ×∇q θ ∇F + ∇ Fx, θ dx , Note that this second order expected estimation 2 0 error cθ[ln(p/c)]2/(2n) for the exponential distribution ˆ ˆ ˆ ˆ where ∇F ∇F(q˜(θ), θ) and ∇2F(x, θ) is the Hessian of is positive. We also note that, all else equal, the second ( , θˆ ) order expected estimation error is monotonically in- F evaluated at x , which are also available without θ knowing the true value of θ. This leads to the fol- creasing in p, is proportional to expected demand , lowing theorem, one of the main results of our paper. and is inversely proportional to n.Asc changes, all else equal, the behavior depends on the sign of the ˆ Theorem 1. If q˜(θ) is smooth as defined at the start of partial derivative Section 3.2, then, in the case of a k-dimensional, k ≥ 1, vector ()[]() ()[]() θ θˆ ∂ θ / 2 θ / / − parameter with unbiased estimator , the adjusted ex- c ln p c ln p c ln p c 2 , pected profit eliminates the O(1/n) bias and is unbiased up to ∂c 2n 2n o(1/n). That is, () < −2 ≈ / . []()() []()() which is positive when c pe p 7 389. Thus, the ˆ ˜ ˆ ˆ ˜ ˆ 1 E πa q θ , θ E π q θ ,θ + o . second order expected estimation error initially increases n with c,reachingitsmaximumvalueof2pθ/(ne2) when −2 3.3. Illustrative Examples c pe , and then decreases to zero when c p with q qˆ 0. In particular, whenever c > p/7.389, the In this section, we apply the general results derived in second order expected estimation error is decreasing the previous sections to representative distributions with c. Finally, replacing the unknown θ value in the and the classic order quantity qˆ to learn how the

second order expected estimation error expression expected estimation error depends on specificsitua- ˆ with its known unbiased estimator θ,wecandeter- tions. We consider the exponential and normal dis- mine the adjusted expected profit. tributions. In particular, we show that the second- order expected estimation error is positive in all cases. Corollary 5. The adjusted expected profit for an exponential distribution is 3.3.1. Exponential Distribution. We first consider the []() () () θˆ / 2 exponential distribution that depends on a single pa- πˆ ˆ, θˆ θˆ − / − ˆ − c ln p c , θ a q p 1 c p cq rameter: its∑ mean .Wemayusetheunbiasedesti- 2n θˆ 1 n [θˆ ]θ (θˆ )θ2/ mator n i1 Xi,whereE and Var n. θ [πˆ (ˆ, θˆ )] [π(ˆ,θ)] + (1) If were known, the optimal order quantity would be where E a q E q o n . q θ ln(p/c). However, the newsvendor naively uses ˆ θˆ ( / ) q ln p c instead. Applying Equation (2), we may 3.3.2. Normal Distribution. We now consider the nor- fi fi nd the expected pro t with exponentially distrib- mal distribution, which is parameterized by the two- ˜ uted demand for an arbitrary order quantity q and vector θ (μ, σ),whereμ is the mean and σ is the θ π(˜,θ) θ( − −q˜/θ)− ˜ mean : q p 1 e cq. This allows us to standard deviation of demand. If μ and σ are known, ˆ π(ˆ, θˆ )−π(ˆ,θ) − determine the estimation error for q, q q q q(θ)μ + σΦ 1( − c) ˆ −ˆ/θˆ −ˆ/θ the optimal order quantity is 1 p . pθ(1 − e q )−pθ(1 − e q ), whose expected value with ˆ The estimated order quantity, based on an estimate respect to the sampling distribution of θ is provided in θˆ (μ,ˆ σˆ) θ (μ, σ) ˆ (θˆ )μˆ + σˆΦ−1( − c) for ,isq q ∑ 1 p . the following proposition. μˆ 1 n We use the arithmetic mean n i1 Xi as an μ Proposition 5. The (positive) profit expected estimation unbiased estimator for . However, the usual esti- σ error for qˆ for the case of an exponential distribution with mators of the standard deviation are known to be mean θ and computed using the general distribution result biased (these usual estimators include the sample of Corollary 2 equals standard deviation that divides the sum of squared − []() () residuals by n 1 before the square root is taken and []() () 2 ˆ cθ ln p/c 1 the MLE that divides by n). Note that the sample E π qˆ, θ − π qˆ,θ + o . σ2 2n n variance is an unbiased estimator of butthatthe nonlinearity of the square root function introduces We may contrast the result of Proposition 5,obtained bias into the sample standard deviation as an esti- by applying the general result of Corollary 3 to the mator of σ,whichisusedtofind the order quantity. Siegel and Wagner: Profit Estimation Error in the Newsvendor Model 4874 Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS

Using properties of the chi distribution (the square Note that the second order expected estimation error root of a chi-squared distribution) with n − 1 degrees for the normal distribution is positive as was the case of freedom from (18.14) of Johnson et al. (1994), we for the exponential distribution. In contrast to the may construct an unbiased estimator σˆ of σ with exponential distribution case, the second order ex- moments as follows: pected estimation error does not depend on the mean √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ μ (although we note a similar proportionate depen- ∑ () 1 n ¯ 2 dence on the standard deviation for both distribu- σˆ ≜ k X − X , n n − 1 i1 i tions). All else equal, the second order expected es- where timation error is proportional to the uncertainty in () demand σ and is inversely proportional to the sample √̅̅̅̅̅̅̅ Γ[]()− / fi fi / √n̅̅ 1 2 size n.Wealso nd that, for a xed ratio c p,the kn ≜ n − 1 2Γ()n/2 second order expected estimation error is propor- () tional to p. Note that the second order expected es- 1 1 1 − () 1 + + O , timation error is symmetric in ξ ≜ Φ 1(1 − c/p) so that − 1 + O 1 4n n2 1 4()n−1 n2 it remains unchanged if c is changed to p − c,replacing ξ with −ξ. All else equal, the second order expected which implies that E[σˆ]σ and Var(σˆ)σ2(k2 − 1) n estimation error is maximized when the cost is half of σ2 + ( 1 ) 2n O n2 .Theexpansionofkn (and, therefore, of the the price because ξ 0 at this point, and this is the variance) are obtained from (18.15) of Johnson et al. only root of the derivative (1994), who credit Johnson and Welch (1939). The expected profit with normally distributed de- () d 2 3 mand for an arbitrary order quantity q˜ and parame- 2 + ξ ϕ()ξ −ξ ϕ()ξ , dξ ters θ (μ, σ),whereΦ and ϕ denote the standard normal distribution and density, respectively, is given by which changes sign from positive to negative at zero. Cachon and Terwiesch (2012)as () () The second order expected estimation error ap- ()()q˜ − μ q˜ − μ π(q˜,θ) p − c q˜ − p q˜ − μ Φ − pσϕ . (4) proaches zero as the cost c approaches either zero or p. σ σ As c approaches zero, the order quantity q approaches ∞, and the newsvendor’sexpectedprofitapproachespμ ˆ μˆ + σˆΦ−1( − / ) Using this, along with the facts that q 1 c p for which we have an unbiased estimate pμˆ ,and μ + σΦ−1( − / ) and q 1 c p , the estimation error (given c θˆ hence, there is no estimation error in this limit. As the estimate )is approaches p, the order quantity q effectively ap- () () ˆ proaches zero, and the newsvendor’sprofit approaches π qˆ, θ − π qˆ,θ [ () () zero, and hence, there is no estimation error in this ()qˆ − μ ()qˆ − μˆ p qˆ − μ Φ − qˆ − μˆ Φ limit either as the estimates become irrelevant. σ σˆ μ σ () ()] Replacing the unknown and values with their ˆ − μ ˆ − μˆ μˆ σˆ + σϕ q − σˆϕ q known unbiased estimators and , respectively, we σ σˆ can determine the adjusted expected profit. { ()()() ()ˆ q − μ c − c Corollary 6. The adjusted expected profit for a normal p qˆ − μ Φ − σˆ 1 − Φ 1 1 − σ p p distribution with unbiased estimated mean μˆ and standard () []}() σˆ qˆ − μ − c deviation is + σϕ − σˆϕ Φ 1 1 − . () σ p ()()()ˆ − μˆ πˆ ˆ, θˆ − ˆ − ˆ − μˆ Φ q a q p c q p q σˆ The expected value of the estimation error with re- () ˆ []() spect to the sampling distribution of θ is provided in ()σˆ + Φ−1 − / 2 qˆ − μˆ p 2 1 c p the following proposition. − pσˆϕ − σˆ n ()() 4 Proposition 6. The profit expected estimation error for qˆ for − × ϕ Φ 1 1 − c/p μ ()[]()() the case of a normal distribution with mean and stan- − σ μˆ σˆ p − c μˆ + σˆΦ 1 1 − c/p − pσˆ 1 − c/p dard deviation and unbiased estimators and , respec- ()[]() − − tively, equals × Φ 1 1 − c/p − pσˆϕ Φ 1 1 − c/p []() () {}[]() ˆ − 2 E π qˆ, θ − π qˆ,θ pσˆ 2 + Φ 1 1 − c/p []() () − ϕ Φ−1 − / , []() () 1 c p pσ 2 + Φ−1 1 − c/p 2 ()() 4n − 1 ϕ Φ 1 1 − c/p + o . 4n n [πˆ (ˆ, θˆ )] [π(ˆ,θ)] + (1) where E a q E q o n . Siegel and Wagner: Profit Estimation Error in the Newsvendor Model Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS 4875

3.4. Managerial Insights newsvendor and π(qˆ,θ) as the newsvendor actually In this section, we unify and discuss the observations experiences (note that these expressions appear in we made in our analysis of the exponential and Sections 3.3.1 and 3.3.2). Their difference, perceived normal distributions. In both cases, we observe that minus actual expected profit, defines the estimation the expected profit estimation error is inversely pro- error before adjustment. For the estimation error after portional to the sample size; in other words, as the adjustment, we subtract the expected estimation error ˆ sample size increases, the profit bias decreases. Thus, in (which is also a function of θ). We are interested in practice, one way to reduce the expected profitestima- exploring the expected values of these two measures tion error is to collect more samples. using simulation. We also find that the exponential distribution’s To improve the efficiency of the Monte Carlo simu- profit estimation error is increasing in the mean θ, lations, for each sample, we create an antithetic sam- whereas the normal distribution’s error is increasing ple by replacing each observation in the sample of 25 in σ (and does not depend on the mean μ). However, with its complementary percentile, which preserves noting that the exponential distribution’smeanisalso distributions and expected values while decreasing equal to its standard deviation, we conjecture a sec- the variance of the simulations as a result of nega- ond commonality: as the demand distribution has tive correlation (Hammersley and Morton 1956). This more uncertainty, the profitbiasincreases. method uses the fact that, if F is a cumulative dis- Regarding the economic parameters of the news- tribution function and U is uniformly distributed vendor, we observe some differences that depend on from zero to one, then both F−1(U) and F−1(1 − U) the distribution: the exponential’sprofitbiasisin- follow the same distribution F but are negatively creasing in p, whereas the normal distribution’sde- correlated. In the normal case, the two values are pendence on p is more subtle. In both cases, we ob- simple reflections about the mean. serve that the bias is unimodal in c with a maximum Thus, for each sample of size 25, we compute es- at a distribution-dependent value of c.Wecanargue timation errors (both before and after adjustment) for that this behavior is general: the bias cannot be both samples (original and antithetic). We then av- monotonic in c for fixed p (if there is a bias for any erage the two values (original and antithetic) for each value of c) because the bias approaches zero at the of the measures (before and after adjustment). The endpoints of the interval c ∈(0, p).Toseethis,note result is a pair of unbiased estimates for the error that, for a fixed value of p, the bias approaches zero as (before and after adjustment). ˆ − c → 0 because the order quantity F 1(1 − c/p) tends to We compute the t-statistic (testing against zero infinity, stock-outs become very rare, and expected estimation error) for each measure (before and after profittendstowardpE[X]; if the newsvendor has an adjustment) by repeating this procedure for 10,000 unbiased estimator of mean demand E[X], then there independent samples (note that, although an indi- is no bias (because there is no nonlinearity). The bias vidual sample and its antithetic counterpart are nega- also approaches zero as c → p from below because the tively correlated, by averaging them, we may focus on ˆ − order quantity F 1(1 − c/p) tends to zero, and the independent random variables while preserving the newsvendor’sestimatedprofit tends to agree closely expectation of interest). The result is a pair of t-statistics withthetrueexpectedprofit because they are both for the expected estimation error: one before and one small numbers. On the other hand, the bias is monotonic after adjustment. in c when the ratio c/p is held fixed, which releases c We then repeat this procedure 100 times to obtain from its upper bound. In fact, the bias is proportional 100 pairs of t-statistics, each based on 10,000 paired to c in this case as it is also with respect to p as may be antithetic simulations. In Figure 2, we plot histograms seen from Corollaries 1 and 3 because q remains of these 100 paired t-statistics for the exponential (left) unchanged when we fixtheratio. and normal (right) distributions. For the exponential distribution, we utilize θ 200; for the normal dis- 3.5. Numerical Verification: Exponential and Normal tribution, we utilize μ 200 and σ 65.Inbothcases, Demand Distributions we set p 5andc 3. We observe strong evidence Simulations were performed for the exponential and that our asymptotic adjustment eliminates the sta- normal demand cases, and considerable expected tistically significant estimation error very effectively estimation error reduction was observed using our even in these finitesamples.IntheleftplotofFigure2, second order approach. Our experimental primitive is the unadjusted estimation error for the exponential as follows: we generate a sample of n 25 observa- distribution shows high statistical significance (i.e., tions from the true distribution (as specified by θ)to the t-values in the histogram at right are considerably ˆ obtain the estimate θ, from which we obtain the order higher than the standard 1.96 critical value), which is size qˆ. We then compute two exact expectations (given successfully eliminated by the asymptotic correc- ˆ this qˆ)ofprofit: π(qˆ, θ) as perceived by the naive tion (i.e., the histogram at left is centered at zero). Siegel and Wagner: Profit Estimation Error in the Newsvendor Model 4876 Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS

Figure 2. (Color online) Adjustment Eliminates Bias for Exponential and Normal Demand

Notes. (Left) Exponential distribution with θ 200, p 5, and c 3 with samples of size n 25. (Right) Normal distribution with μ 200, σ 65, p 5, and c 3 with samples of size n 25.

(∇ )(∇ ) − (∇ )(∇ ) + 2(∇2 ) fi γ ≜ − [ −1(θ)( fq F F 2f F f f F )] For exponential demand, the expected pro t is 90.4, where tr I 2f 3 .The and the expected estimation error is 3.1, which is a expected estimation error in the order quantity for the case nonnegligible 3.4% of the profit. In the right plot of of a one-dimensional parameter θ with unbiased estima- ˆ Figure 2, we observe similar behavior for the normal tor θ equals distribution for which the expected profit is 271.9 and () [] () 2 − + 2 the expected estimation error is 2.6, which is 1% and ˆ fqFθ 2fFθf θ f Fθθ 1 E qˆ − q −Var θ + o . less than in the exponential case but still nonnegligible. 2f 3 n This smaller expected estimation error in the normal case may be due to its smaller standard deviation. 4.1. Unbiased Adjusted Order Quantity Estimation In this section, we define an adjusted order quantity, 4. The Impact of Parameter Estimation on ˆ the Order Quantity denoted as qa, which provides an unbiased estimate of the true optimal order quantity. To construct this

In the previous section, we show that, despite uti- adjusted order quantity, we modify the newsvendor’s lizing unbiased estimates of the parameters of the ˆ naive order quantity q(θ) by subtracting a consistent probability distribution of demand, the newsvendor estimate of the second order expected estimation profit exhibits a nontrivial estimation error. In this error term identified in Proposition 7 (substituting the section, we study whether the order quantity itself ˆ consistent estimators θ for θ and qˆ for q,inparticular has an estimation error and show how to correct this to estimate the partial derivatives). error when it exists. As in the previous section and using Equation (1), Proposition 8. In the case of a k-dimensional parameter θ ˆ we may define the estimation error as the difference with unbiased estimator θ,ifwedefine the adjusted order ˆ between naive and optimal order quantities, q(θ)− quantity as ˆ q(θ)q − q, which is again a random variable as it ⎛ ⎞ ˆ ⎡⎢ ˆ ˆ ˆ ˆ ˆ ⎤⎥ depends on the random variable θ.Averagingover ⎢ f ()∇F ()∇F −2f ()∇F ⎥ ˆ ⎢ ⎜ q ⎟⎥ θ ⎢ ⎜ ˆ ˆ2 ˆ ⎟⎥ the sampling distribution of , we obtain the expected ⎢ ()⎜ × ()∇f + ()∇2F ⎟⎥ ˆ ≜ ˆ + 1 ⎢ −1 θˆ ⎜ f ⎟⎥ qa q tr⎢ I ⎜ ⎟⎥ estimation error ⎢ ⎜ ˆ3 ⎟⎥ n ⎢ ⎝⎜ 2f ⎠⎟⎥ [] ⎣⎢ ⎦⎥ E qˆ − q . γˆ qˆ − , Using similar techniques as in the previous section, n weareabletoprovethefollowingresult. then E[qˆ ]q + o(1) so that the O(1/n) term has been Proposition 7. a n The expected estimation error in the order removed from the bias of q,ˆ where our consistent estima- quantity for the case of a k-dimensional parameter θ ˆ tor is (θ , ...,θ ) with unbiased estimator θ equals 1 k () () ⎛ ()() () ()⎞ ⎛ ⎞ ⎢⎡ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ⎥⎤ ⎢⎡ ()∇ ()∇ − ()∇ ∇ ⎥⎤ ⎢ ()⎜f ∇F ∇F − 2f ∇F ∇f + f 2 ∇2F ⎟⎥ ⎢ ⎜fq F F ()2f F f ⎟⎥ ⎢ − ⎜ q ⎟⎥ ⎢ ⎜ ⎟⎥ γˆ ≜ − ⎢ 1 θˆ ⎜ ⎟⎥. [] ⎢ ()⎜ + f 2 ∇2F ⎟⎥ tr⎢ I ⎝ ˆ ⎠⎥ ˆ − − ⎢ θˆ ⎜ ⎟⎥ ⎣⎢ 2f 3 ⎦⎥ E q q tr⎢Cov ⎜ ⎟⎥ ⎢ ⎝⎜ 2f 3 ⎠⎟⎥ ⎣⎢ ⎦⎥ () () We next show that the order-quantity estimation 1 γ 1 fi + o + o , error does not materially affect the expected-pro t n n n estimation error. That is, the naively estimated order Siegel and Wagner: Profit Estimation Error in the Newsvendor Model Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS 4877 quantity can itself have an estimation error, which can Proposition 10. For the log-normal distribution, the as- be corrected via Proposition 8; however, the second ymptotic estimation error-correction term for the estimated order profit expected estimation error formula re- order quantity is mains unchanged (from its form in Corollary 1), and () γ σ2 + ξ2 the expected profit is not materially improved by the 2 q , estimation error correction of the estimated order n 4n quantity (i.e., this change is zero to second order). We ˆ ˆ ˆ so the adjusted estimated order quantity is qa q − γ/n, work with the general case of k ≥ 1 real parameters in σˆ 2( +ξ2)ˆ − where γˆ 2 q and ξ Φ 1(1 − c/p). the vector θ. We next show that the second-order 4 profit expected estimation error for the adjusted or- The next result provides the profitexpectedesti- der quantity is the same as that of the unadjusted mation error when demand is log-normally distrib- order quantity. uted; note that, per Corollary 1 and Proposition 9,the expressionisthesamewhenusingtheunadjusted Proposition 9. The profit expected estimation error E[π biased order quantity. ˆ ˆ ˆ ˆ (qa, θ)−π(qa,θ)] for qa for the case of a k-dimensional ˆ Proposition 11. For the log-normal distribution, the profit parameter θ (θ , ...,θ ) with unbiased estimator θ ˆ 1 k expected estimation error E[π(qˆ , θ)−π(qˆ ,θ)] for qˆ equals equals a a a []()∫ () pσ [ ()() ()()∇ ()∇ q q 2 + ξ2 − σξ − σ2 ϕ()ξ + σ 3 + σ2 · θˆ F F − 1 ∇2 (),θ + 1 p tr Cov Fx dx o 4n ] () f 2 0 n []()∫ () × μ+σ2/2Φ ξ − σ + 1 . q e ()o p − ()∇F ()∇F 1 1 tr I 1()θ − ∇2Fx(),θ dx + o . n n f 2 0 n Furthermore, the adjusted expected profit for a log-normal ˆ ˆ fi distribution with unbiased estimated parameters μ and σ is Furthermore, the change in actual expected pro t(un- ()() der the true demand parameter θ) is negligible; that is, E[π ()() ˆ − μˆ ˆ ˆ ˆ ˆ ˆ ln qa (ˆ ,θ)−π(ˆ,θ)] (1) πa qa, θ p − c qa − pqaΦ qa q o n . σˆ ()() We recognize that the firstresultinProposition9 ˆ ˆ ˆ 2 2 q − μ − σ + μˆ +σˆ /2Φ ln a has the same functional form as that in Corollary 1, pe σˆ which is why the second result shows that the order σˆ [ () fi p ˆ 2 ˆ ˆ 2 quantity adjustment results in no additional pro tto − qa 2 + ξ − σξ − σ ϕ()ξ second order. 4()n ] μˆ +σˆ 2/ + σˆ 3 + σˆ 2 e 2Φ()ξ − σˆ . 4.2. Illustrative Example: Log-Normal Demand Distribution We now study the log-normal demand distribution as 4.3. Numerical Verification: Log-Normal an example with a biased order quantity. Note that Demand Distribution the exponential distribution has an unbiased order As in Section 3.5, we perform numerical experiments ˆ quantity because E[qˆ]E[θln(p/c)]θln(p/c)q. Simi- that provide statistical evidence of nontrivial profit larly, for normally distributed demand with the estimation error and now also order-quantity esti- unbiased estimates μˆ and σˆ,wefind E[qˆ]E[μˆ + mation error for the log-normal distribution (a heavy- − − σˆΦ 1(1 − c/p)] μ + σΦ 1(1 − c/p)q. This generalizes tailed distribution; see Foss et al. 2013). Demand is to show that the order quantity is unbiased for any simulated from a log-normal distribution with mean location-scale family of demand distributions when 200 and standard deviation 65 so that, on the log scale, unbiased parameter estimates are used. we have θ (μ, σ)(5.248, 0.317).Theremainderof Log-normal demand may be represented as X eY, the experimental setup is as in Section 3.5.Wefind where Y (the log of demand) has a normal distribution that the expected profitis282.1,andtheexpected with mean μ and standard deviation σ.Theunbiased estimation error is 3.1. Furthermore, in the left plot of mean and standard deviation estimates (μˆ and σˆ as Figure 3 (which uses the bias-adjusted order quantity used in the normal case) may now be computed using while exploring profit bias), we again observe strong the logs of sampled demands. The unadjusted order statistical evidence of the nonzero profitexpected quantity for the log-normal distribution is, there- estimation error: the unadjusted expected estimation ˆ μˆ +σˆΦ−1( − / ) fore, qˆ q(θ)e 1 c p . error shows high statistical significance (t-values in Siegel and Wagner: Profit Estimation Error in the Newsvendor Model 4878 Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS

Figure 3. (Color online) Log-Normal Distribution with Mean 200, Standard Deviation 65, p 5, and c 3 with Samples of Size n 25

ˆ Note. The left plot, which presents profit estimation errors, is for the adjusted order quantity qa. the histogram at right are considerably higher than this work to the case of sales data (i.e., censored de- the 1.96 critical value), which is successfully elimi- mand data). Finally, Although we focused on ana- nated by the asymptotic correction (histogram at left lyzing the impact of estimation error for the news- is centered at zero). vendor model, we suspect many parametric approaches In the right plot of Figure 3, we observe similar to optimization under uncertainty that use real data for evidence for the order quantity: expected estimation estimation are significantly affected by estimation error; error in the unadjusted order quantity qˆ shows high it might be possible to derive closed-form adjustment statistical significance (t-values in the histogram at terms in other contexts to provide unbiased estimates of right are considerably higher than the 1.96 critical meaningful quantities. value), which is successfully eliminated by the as- ˆ ymptotic correction for the order quantity qa (histo- Acknowledgments gram at left is centered at zero). The naive news- The authors thank the department editor, associate editor, vendor orders, on average, approximately 0.36 to 0.37 and three anonymous referees for very useful feedback that more than the true optimal order quantity q 175.534. greatly improved the quality of the paper. This does not materially affect the true expected profit as is consistent with Proposition 9: using the unad- References justed order quantity qˆ reduces true expected profitby APICS (2019) APICS dictionary. Accessed November 1, 2019, http:// only an estimated 0.003 from the expected profites- www.apics.org/apics-for-individuals/publications-and-research/ timated as 282.1 (0.001%) as compared with the apics-dictionary. material expected estimation error of 3.1 (1.1%) as- Arnold B, Balakrishnan N, Nagaraja H (2008) A First Course in Or- fi θˆ der Statistics (Society for Industrial and Applied Mathematics, sociated with naively computing the pro tusing in Philadelphia). place of the true demand parameter θ. Ban G, Rudin C (2019) The big data newsvendor: Practical insights from machine learning. Oper. Res. 67(1):90–108. Ban G, Gallien J, Mersereau A (2019) Dynamic procurement of new 5. Conclusion products with covariate information: The residual tree method. In this paper, in the context of the newsvendor model, Manufacturing Service Oper. Management 21(4):798–815. we show that unbiased estimators of distributional Bentzley J (2017) The overlooked forecasting flaw: Forecast bias and how to tackle it. Accessed November 1, 2019, https://www parameters, when passed through an optimization fl operator, result in biased estimates of the optimal .linkedin.com/pulse/overlooked-forecasting- aw-forecast-bias -how-tackle-jim-bentzley. objective function value and, depending on the de- Besbes O, Muharremoglu A (2013) On implications of demand mand distribution, biased order quantities. We derive censoring in the newsvendor problem. Management Sci. 59(6): closed-form second order approximations for the 1407–1424. expected estimation errors and use these to form Burnetas A, Smith C (2000) Adaptive ordering and pricing for per- – adjusted expected-profit and order-quantity expres- ishable products. Oper. Res. 48(3):436 443. Cachon G, Terwiesch C (2012) Matching Supply with Demand: An sions that can be computed by a newsvendor who Introduction to Operations Management, 3rd ed. (McGraw-Hill, does not know the true demand distribution. We New York). conduct simulation studies for exponential, normal, Cassar G, Gibson B (2008) Budgets, internal reports, and manager and log-normal demand distribution families and forecast accuracy. Contemporary Accounting Res. 25(3):707–737. find statistically significant estimation errors in each Chu L, Shanthikumar J, Shen Z (2008) Solving operational statistics via a Bayesian analysis. Oper. Res. Lett. 36(1):110–116. case that are effectively eliminated by our adjustments. Durand R (2003) Predicting a firm’s forecasting ability: The roles of We conclude by commenting on further research. It organizational illusion of control and organizational attention. would be interesting and practically useful to extend Strategic Management J. 24(9):821–838. Siegel and Wagner: Profit Estimation Error in the Newsvendor Model Management Science, 2021, vol. 67, no. 8, pp. 4863–4879, © 2020 INFORMS 4879

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