Convex-transform comparisons beyond common-shape component families

Determine whether convex-transform ordering results continue to hold when component distributions have different shape or tail parameters, and characterize the conditions on a baseline cumulative hazard that are sufficient to preserve the comparison.

Background

The paper establishes convex-transform comparisons between heterogeneous and homogeneous order statistics when all components share a common baseline family and differ only through their rate parameters. For the proportional-hazards family, the comparison is proved under the shape restriction 0<α≤1, with a rate condition required when ξ>0; the paper also shows that the ordering fails in general for Weibull distributions with α>1.

The authors explicitly identify as unresolved whether analogous comparisons remain valid when components have different shape or tail parameters. They further ask for a characterization of baseline cumulative-hazard conditions that would preserve the ordering, making this a broader generalization of the paper’s common-baseline results.