On the Entropy of Compound Distributions on Nonnegative Integers

Negative dependence and stochastic orderings

“…Since this is stronger than stochastic ordering [25, Theorem 1.C.1], our Corollary 2.3 strengthens Theorem 2.5 of [16]. Similarly, Corollary 2.4 below generalises Theorem 3 of [29]. See also [17].…”

“…Proof. Since W ≤ cx Z λ (by Theorem 2.2) and Z λ is a log-concave random variable, the result follows from Lemma 1 of [29].…”

“…Daly et al [8] give bounds on the Poisson approximation of W in total variation distance under the assumption that [16] and [29].…”

Entropy and Thinning of Discrete Random Variables

“…4.4. A related extension was given in the compound Poisson case in[59], one assumption of which was removed by Yu[105]. Yu's result[105, Theorem 3] showed that (assuming it is log-concave) the compound Poisson distribution CP (λ, Q) is maximum entropy among all distributions with 'claim number distribution' in ULC(λ) and given…”

On an inequality of Karlin and Rinott concerning weighted sums of i.i.d. random variables

“…For example, if X i ∼ Gam(1/n, 1), i.e., a gamma distribution with shape parameter 1/n, then the equally weighted n i=1 X i , which has an exponential distribution, maximizes rather than minimizes the entropy H among n i=1 a i X i with n i=1 a i = n. For more entropy comparison results where log-concavity plays a role, see Yu (2009aYu ( , 2009b. Karlin and Rinott (1981) conjectured Theorem 1 (their Remark 3.1, p. 110) and proved a special case (their Theorem 3.1) assuming that i) a i > 0 and ii) f (x), the density of the X i 's, is supported on [0, ∞), and admits a Laplace transform of the form…”