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For nonnegative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the 'expected reversed proportional shortfall order', and a new characterization of random lifetimes involving the reversed hazard rate function.Short title: A quantile-based probabilistic mean value theorem.

a b s t r a c tDifferent sufficient conditions for stochastic comparisons between random vectors have been described in the literature. In particular, conditions for the comparison of random vectors having the same copula, i.e., the same dependence structure, may be found in Müller and Scarsini (2001). Here we provide conditions for the comparison, in the usual stochastic order sense and in other weaker stochastic orders, of two time transformed exponential bivariate lifetimes having different copulas. Some examples of applications are provided too.

In this paper, we provide a new concept of relative skewness among multivariate distributions, extending to the multivariate case a similar concept in the univariate case. In this case, a random variable Y is said to be more right skewed than a random variable X if there exits an increasing convex transformation which maps X onto Y .Given two random vectors X and Y and an appropriate transformation which maps X onto Y, we define a new concept of relative skewness assuming the convexity of this transformation. Properties and applications of this concept are given.