Sufficient Conditions for Some Stochastic Orders of Discrete Random Variables with Applications in Reliability

In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio for the comparison in some stochastic orders of two discrete random variables. These results have interest in comparing discrete random variables because the sufficient conditions are easy to check when there are no closed expressions for the survival functions, which occurs in many cases. In addition, the results are applied to compare several parametric families of discrete distributions.

💡 Research Summary

The paper addresses a notable gap in reliability theory: the lack of sufficient conditions for stochastic ordering of discrete random variables. While stochastic orders such as the usual stochastic order (st), hazard‑rate order (hr), mean‑residual‑life order (mrl), and likelihood‑ratio order (lr) are well‑studied for continuous lifetimes, their discrete counterparts have received little attention. The authors focus on the hazard‑rate and mean‑residual‑life orders, which are particularly important in reliability and survival analysis, and they propose easily verifiable conditions that do not require closed‑form survival functions.

Key concepts and definitions

• For a non‑negative integer‑valued random variable (X) the mass function (f_X), survival function (S_X(x)=P(X\ge x)), hazard rate (h_X(x)=P(X=x\mid X\ge x)=f_X(x)/S_X(x)), and mean residual life (m_X(x)=E