  We consider the construction of insurance premiums that are monotonically
increasing with respect to a loading parameter. By introducing weight functions
that are totally positive of higher order, we derive higher monotonicity
properties of weighted transformed premiums. We deduce that the greater the
degree of randomness of insured risks, the higher the order of total positivity
that should be required for the chosen weight functions. We examine seven
classes of weight functions that have appeared in the literature, and we
ascertain the higher order total positivity properties of those functions.
