Recall, from Subsection 2.16.4, the likelihood ratio statistic, Ln, which was defined as a product of independent, identically distributed random variables with mean 1 (under the so-called null hypothesis), and the, sometimes more convenient, log-likelihood, log Ln, which was a sum of independent, identically distributed random variables, which, however, do not have mean log 1 = 0.
(a) Verify that the last claim is correct, by proving the more general statement, namely that, if Y is a non-negative random variable with finite mean, then
E(logY) < log(EY).
(b) Prove that, in fact, there is strict inequality:
E(logY)
unless Y is degenerate.
(c) Review the proof of Jensen's inequality. Generalize with a glimpse on (b).