Stefanski (1996) establishes the arithmetic-geometric-harmonic mean inequality (see Example 4.7.8 and Miscellanea 4.9.2) using a proof based on likelihood ratio tests. Suppose that Y1,...,Yn are independent with pdfs λie-λiyi', and we want to test H0: λ1 = . . . = λn vs. H1: λi, are not all equal.
(a) Show that the LRT statistic is given by ()"n/(Πi ,Yi)-1 and hence deduce the arithmetic-geometric mean inequality.
(b) Make the transformation X1 = 1 /Yi, and show that the LRT statistic based on
X1,..., Xn is given by [n/∑i(1/Xi)]n/Π Xi and hence deduce the geometric-harmonic mean inequality.