Further Research Directions

Abstract

In the monograph we mostly focused on the optimal upper bounds, but certainly the lower ones are needed for evaluating the actual ranges of the functionals over given classes of distributions. Generally, the best upper and lower bounds are not symmetric about zero, but the latter can also be derived by means of our projection method. To this end we should analyze the negatives of the functionals under study. Only the functional corresponding to the order statistics and L-statistics (see (2.27) and (2.25)) in the dependent case needs a more subtle transformation. Changing the signs of coefficients c _j , 1 ≤ j ≤ n, in (2.26) results in constructing a functional g−c different from −g_c. Also, one should realize that generally projecting a functional and its negative are different problems that should be solved separately by use of specific arguments. With few exceptions they cannot be derived one from the other in a simple way.