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The Stieltjes transform order and related ratio order

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  • Alenazi, Abdulaziz
  • Mehrez, Khaled

Abstract

In this paper, we introduce the Stieltjes transform order, defined via the composition of the Laplace transform of a random variable. First, we establish the monotonicity of the hazard rate associated with a transformed distribution of a random variable X. As a consequence, we derive a sharp lower bound for the Stieltjes transform of X in terms of its first two inverse moments. Secondly, we investigate the Stieltjes transform ratio order and provide its characterizations, both via properties of the hazard rate and through the monotonicity properties of higher-order Stieltjes transforms together with moment inequalities. We further establish explicit connections between the proposed orders and classical stochastic orders, including equivalence with the usual stochastic order and implication from the likelihood ratio order. These results enhance our understanding of stochastic orders and open new avenues for applications in reliability theory, economics, and applied probability.

Suggested Citation

  • Alenazi, Abdulaziz & Mehrez, Khaled, 2026. "The Stieltjes transform order and related ratio order," Statistics & Probability Letters, Elsevier, vol. 233(C).
    • Handle: RePEc:eee:stapro:v:233:y:2026:i:c:s0167715226000271
    DOI: 10.1016/j.spl.2026.110663
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