Refer to Miscellanea 4.9.2.
(a) Show that Ai is the arithmetic mean, A-1 is the harmonic mean, and A0 = limr†’0 Ar is the geometric mean.
(b) The arithmetic-geometric-harmonic mean inequality will follow if it can be established that Ar is a nondecreasing function of r over the range -ˆž (i) Verify that if log Ar is nondecreasing in r, then it will follow that Ar is non-decreasing in r.
(ii) Show that

(iii) Define ai = xri/ˆ‘i xri and write the quantity in braces as

where ˆ‘ai = 1. Now prove that this quantity is nonnegative, establishing the monotonicity of Ar and the arithmetic-geometric-harmonic mean inequality as a special case.
The quantity ˆ‘i ai, log(l/ai) is called entropy, sometimes considered an absolute measure of uncertainty (see Bernardo and Smith 1994, Section 2.7). The result of part (iii) states that the maximum entropy is attained when all probabilities are the same (randomness).
(To prove the inequality note that the ai are a probability distribution, and we can write

and Jensen's Inequality shows that E log (1/a)

Transcribed Image Text:

108 Ar dr Σ a, log(1/a), l og n) - ai