Estimation of quantiles of non-stationary demand distributions

Many problems involve the use of quantiles of the probability distributions of the problem's parameters. A well-known example is the newsvendor problem, where the optimal order quantity equals a quantile of the demand distribution function. In real-life situations, however, the demand distribution is usually unknown and has to be estimated from past data. In these cases, quantile prediction is a complicated task, given that (i) the number of available samples is usually small and (ii) the demand distribution is not necessarily stationary. In some cases the distribution type can be meaningfully presumed, whereas the parameters of the distribution remain unknown. This article suggests a new method for estimating a quantile at a future time period. The method attaches weights to the available samples based on their chronological order and then, similar to the sample quantile method, it sets the estimator at the sample that reaches the desired quantile value. The method looks for the weights that minimize the expected absolute error of the estimator. A method for determining optimal weights in both stationary and non-stationary settings of the problem is developed. The applicability of the method is illustrated by solving a problem that has limited information regarding the distribution parameters and stationarity.