Preservation of properties under mixture

In general, finite mixtures of distributions with increasing failure rates are not increasing. However, conditions have been given by Lynch [8] so that a mixture of distributions with increasing failure rates has increasing failure rate. We establish similar results for other standard classes and also give examples which show that although the assumptions are stringent, continuous mixtures of standard families of lifetime distributions do have increasing failure rates. We also show that the result of Lynch follows from Savits [12] and the techniques of the last-cited paper can be applied to other classes as well.