A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables

Abstract

Let {X i, 1≤i≤n} be a negatively associated sequence, and let {X* i , 1≤i≤n} be a sequence of independent random variables such that X* i and X i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef(∑ n i=1 X i)≤Ef(∑ n i=1 X* i ) for any convex function f on R 1 and that Ef(max1≤k≤n ∑ n i=k X i)≤Ef(max1≤k≤n ∑ k i=1 X* i ) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.