Use of Uncertain External Information in Statistical Estimation

: A product’s life cycle hinges on its sales. Product sales are determined by a combination of market demand, industrial production, logistics, supply chains, labor hours, and countless other factors. Business-specific questions about sales are often formalized into questions relating to specific quantities in sales data. Statistical estimation of these quantities of interest is crucial but restricted availability of empirical data reduces the accuracy of such estimation. For example, under certain regularity conditions the variance of maximum likelihood estimators cannot be asymptotically lower than the Cramer-Rao lower bound. The presence of additional information from external sources therefore allows the improvement of statistical estimation. Two types of additional information are considered in this work: unbiased and possibly biased. In order to incorporate these two types of additional information in statistical estimation, this manuscript minimizes mean squared error and variance. Publicly available Walmart sales data from 45 stores across 2010-2012 is used to illustrate how these statistical methods can be applied to use additional information for estimating weekly sales. The holiday effect (sales spikes during holiday weeks) adjusted for overtime trends is estimated with the use of relevant external information.