DUAL POINCARÉ-TYPE INEQUALITIES VIA THE CRAMÉR-RAO AND THE CAUCHY-SCHWARZ INEQUALITIES AND RELATED CHARACTERIZATIONS

Abstract

Upper and lower bounds for the variance of a function g(X) of a random vector X(X1,…,Xn) are revisitied. Under independence of the Xi, an elementary derivation of the inequalities (1.1) and (1.4) is given, exhibiting also the nature of their duality, in the sense that replacing E and g21 in the upper bound by E2 and gi, respectively, gives the lower bound. The same functions wixi) appear in both bounds and they are shown to characterize the distribution of X. Equality is attained in both bounds iff g is linear.