On the problem of measuring the distance between distribution functions: analysis of hospitalization versus mortality

The relationship between hospitalization, as a measure of morbidity, and mortality is examined. The difference between age at hospitalization in a general medical department and age at death in Oslo, Norway, is studied. The problem is transferred to the one of examining the difference between two cumulative distribution functions F and G. For this purpose, a quantile distance function based on the inverses of the distribution functions is applied. We give the natural estimate of the quantile distance function, and some asymptotic properties of the corresponding empirical process. For the particular situation where one of the distribution functions is known, a confidence band for the quantile distance function is derived. Applying these results, we show that there are reasons to believe that age at hospitalization and age at death are equally distributed, apart from a constant shift.