General Distributional Properties of Discounted Warranty Costs With Risk Adjustment Under Minimal Repair

We study the distributional properties (mean, variance, characteristic function) of the discounted warranty cost (DWC) for general warranty programs including free replacement (FRW), pro rata (PRW), and FRW/PRW in the context of minimal repair. Because only failure times & types are needed in the derivation of these properties, the reliability of the systems is modeled according to a general competing risk model. Under these assumptions, our results extend those obtained in 2004 by Bai & Pham. By obtaining the characteristic function of the DWC, we can consider some risk management issues, an area that has not yet been extensively studied in this context. More precisely, we show how risk adjustment principles considered in the economics and actuarial science literature can be applied to the determination of a warranty reserve. Because some of these risk adjustment calculations require the probability mass or density function of the DWC, we show how to numerically invert the characteristic function of the DWC to obtain risk adjusted warranty costs with MATLAB in an appendix.