Conditional m-spacings demonstrate log-convexity and log-concavity in stochastic models, indicating their theoretical implications.
In this paper, the concepts of conditional m-spacings are introduced when a series system with independent and identically distributed (i.i.d.) components fails at a given time t. First, we prove that the conditional spacings Di,n−11,t preserves the log-convexity of the parent density, while the conditional m-spacings Di,n−1m,t(m≥2) preserve its log-concavity. Second, we extend the likelihood ratio order and hazard rate order from m-spacings to conditional m-spacings. In addition, we present some examples involving Pareto and Gamma distributions.
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Zhang et al. (2026) studied this question.
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