Stochastic Comparisons of Weighted Sums of Random Variables

Let X = (X 1, …, X n) be a random vector of observations which may not be independent or identically distributed, and let T = ∑ i = 1 n θ i X i , $$\displaystyle T = \sum _{i=1}^n \theta _i X_i, $$ be a linear function of X 1, …, X n, where θ i, i = 1, …, n, are constants. In this chapter we stochastically compare statistics of the above type as the coefficients θ i’s vary. The theory of majorization is used to find conditions on the vector of θ’s under which the weighted sums of X i’s are ordered according to various stochastic orders like the likelihood ratio, the hazard rate, the usual stochastic, the peakedness, the dispersive, and the right spread orders. The case of weighted sums of gamma distributions is studied in detail. The convolutions of Bernoulli and geometric random variables are stochastically compared as their parameters vary. The topic of weighted sums of dependent random variables is also considered.