Loading Monotonicity of Weighted Premiums, and Total Positivity Properties of Weight Functions

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.