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

Abstract

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 and left-continuous version of the survival function We determine the points, if any, such that and We offer an alternative proof of Markov's inequality by observing that, if some collection of events exists such that then because equals a probability, it must satisfy 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 of n non negative observations 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).