Skewness and Relative Aging Orders

A random variable X is said to have symmetric distribution (about a constant c) if the random variables X − c and c − X have the same distribution. The notion of skewness is intended to represent departure of a density from symmetry where one tail of the density is stretched out more than the other. The concept of skewness has been applied particularly to unimodal densities with support on [0, ∞). If the mode of a distribution is to the left of the center and the right-hand tail is relatively thin and long, then the density is said to be skewed to the right. Skewed data is observed in many areas such as economics, reliability, survival analysis, insurance, and social sciences. Lifetime distributions are typically skewed. The concept of positive aging describes the adverse effects of age on the lifetime of units or individuals. Various aspects of this concept are described in terms of conditional distributions of residual lifetimes, failure rates, mean residual lifetimes, etc. Negative exponential distribution is the only distribution for which aging has no effect. The lifetimes are usually positively skewed. It is also of interest to compare the relative aging of two or more distributions. In this chapter various partial orders which compare the relative skewness and aging between probability distributions are discussed. These include the convex-transform order, the star order, the super-additive order, the more DMRL order, the more NBUE order, and the Lorenz order. Conditions under which skewness orders imply variability orders are investigated. Some statistical inference procedures associated with these orders are also discussed.