Some New Results Related to the Likelihood Ratio and the Laplace Transform Ordering

The purpose of this paper is twofold. First, we aim to study some new properties and relations for the likelihood ratio order. In particular, we present new integral inequalities involving the ratio of two probability density functions and the ratio of two corresponding cumulative distribution functions. Specifically, we provide alternative proofs for certain relationships between the likelihood ratio order and other stochastic orders. Second, we consider the Laplace ratio order and present new characterizations for the Laplace ratio ordering of two random variables. These results contribute to the literature on stochastic orders by providing new characterizations and inequalities that can be useful in probability theory, risk analysis and reliability theory.