LAUSR

SYNOPTIC ABSTRACT This article opens up new aging classes and new stochastic orders dependant on generalized moment-generating function, which plays a vital role in reliability theory, finance topics, stochastic orders, and economic theory. The article presents some important new implications concerning these classes and introduces characterizations of the Weibull distribution, exponential distribution, and Erlang distribution through newly suggested aging classes. In addition, we list a series of inequalities that provide bounds for generalized moment-generating classes. Furthermore, a sufficient condition for allowing a probability distribution to have a new class is provided. The article also demonstrates the preservation properties of a new stochastic order under certain reliability operations, such as mixture and convolution. Moreover, some novel applications of classes and stochastic orders in random sum and shock models are provided.