Abstract: The properties of the log-density ratio function are utilized to analyze the hazard rate order relationship for general non-negative random variables under conditions weaker than the concave function.Then,by leveraging the properties of density generator functions,the influence of shape parameters on the magnitudes of the density functions of folded ellipsoidal random variables is analyzed.Furthermore,an equivalent characterization is provided for the stochastic comparison of hazard rate order for folded ellipsoidal random variables under different density generator functions and conditions weaker than the concave function.

Key words: folded elliptical distribution family, hazard rate order, logarithmic density ratio function, density generator function