Estimating the cumulative distribution function of lead-time demand using bootstrapping with and without replacement

Abstract

Forecasting of the cumulative distribution function (CDF) of demand over lead time is a standard requirement for effective inventory replenishment. In practice, while the demand for some items conforms to standard probability distributions, the demand for others does not, thus making it challenging to estimate the CDF of lead-time demand. Distribution-free methods have been proposed, including resampling of demand from previous individual periods of the demand history, often referred to as bootstrapping in the inventory forecasting literature. There has been a lack of theoretical research on this form of resampling. In this paper, we analyse the bias and variance of CDF estimates obtained by resampling, both with and without replacement. Counterintuitively, we find that the ‘with replacement’ approach does not always dominate ‘without replacement’ in terms of mean square error of CDF estimates. Closed-form expressions are given for the components of Mean Square Error, with and without replacement. For shorter lead times, of two or three periods, these may be used directly to identify series that may benefit from resampling without replacement. Inventory performance implications are evaluated on simulated and empirical data. It is found that marked differences may arise between ‘with replacement’ and ‘without replacement’ bootstrapping approaches. The latter method can be more beneficial for lower target Cycle Service Levels, longer lead times and shorter demand histories.