Markov's inequality: Sharpness, renewal theory, finite samples, reliability theory

We examine Markov's inequality in the light of renewal theory and reliability theory. Suppose the non negative random variable (rv) X has cumulative distribution function (cdf) F with survival function F¯≔Pr⁡(X>x)=1−F and left-continuous version of the survival function F¯(x−)≔Pr⁡(X≥x), x≥0. We determine the points, if any, such that x≥μ and F¯(x−)=μ/x. We offer an alternative proof of Markov's inequality by observing that, if some collection of events {At:t≥0} exists such that Pr⁡(At)=tF¯(t−)/μ, then because tF¯(t−)/μ equals a probability, it must satisfy tF¯(t−)/μ≤1, which is equivalent to Markov's inequality. We choose events connected to stationary renewal processes. When we know only the sample size n and the sample average x¯n of n non negative observations x1,…, xn, we establish an upper bound on the left-continuous version of the empirical survival function that improves Markov's inequality. We show that an upper bound of Markov type for the survival function is sharp when F is “new better than used in expectation” (NBUE) or has “decreasing mean residual life” (DMRL).