Stochastic Ordering: from the Lorenz curve to Quality Control

The desire to compare what is random is probably as old as probability itself. A landmark in the history of stochastic ordering (SO) is the pioneering work by Lorenz (1905) in the assessment of income inequality in a population of n individuals. Lorenz - feeling that all of the summary measures then under consideration constituted too much condensation of the data - proposed what is now known as the Lorenz curve and suggested the following rule of interpretation: a high level of income inequality is associated to a severely bent curve. Since this tool was developed by Lorenz, SO has gained widespread acceptance and has been applied, namely, in Biology, Queueing Theory, Reliability Theory, Risk Theory, Scheduling, and Statistical Inference. This presentation will focus on a brief overview of SO and on two applications. In Finance, SO is particularly important in demand and shift effect problems in portfolio selection. As for Quality Control, SO provides decisive insights into how control schemes work in practice and allows the performance comparison of competitive schemes.