Product of independent, Recall, from Subsection 2.16.4, the

Product of independent

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Reference no: EM131830168

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) <log(EY)

unless Y is degenerate.

Reference no: EM131830168

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