Kernel Characterisations of Stochastic Orders Within Parametric Density Families

Abstract

We develop kernel criteria for the likelihood-ratio, hazard-rate, usual stochastic, and relative log-concavity orders in parametric families of univariate probability laws with densities. The score is the derivative of the log density with respect to the parameter, and a kernel equals the score up to an additive term depending only on the parameter. Kernel monotonicity gives likelihood-ratio order, kernel concavity gives relative log-concavity, and two tail-conditional mean inequalities give the hazard-rate and usual stochastic orders. The same construction applies along joint-parameter paths and to comparisons between two laws whose densities admit parameter-dependent factors, where the log-factor ratio is used as the kernel. For compound sums with a random number of i.i.d. terms, the induced kernel is the posterior mean of the kernel of the summand count. The applications recover standard one-parameter orderings, give likelihood-ratio comparisons for compound laws, and handle nonmonotone examples through the tail-conditional criteria.