On the inclusion probabilities in some unequal probability sampling plans without replacement

Comparison results are obtained for the inclusion probabilities in some unequal probability sampling plans without replacement. For either successive sampling or H\'{a}jek's rejective sampling, the larger the sample size, the more uniform the inclusion probabilities in the sense of majorization. In particular, the inclusion probabilities are more uniform than the drawing probabilities. For the same sample size, and given the same set of drawing probabilities, the inclusion probabilities are more uniform for rejective sampling than for successive sampling. This last result confirms a conjecture of H\'{a}jek (Sampling from a Finite Population (1981) Dekker). Results are also presented in terms of the Kullback--Leibler divergence, showing that the inclusion probabilities for successive sampling are more proportional to the drawing probabilities.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ337 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Relative log-concavity and a pair of triangle inequalities

The relative log-concavity ordering $\leq_{\mathrm{lc}}$ between probability mass functions (pmf's) on non-negative integers is studied. Given three pmf's $f,g,h$ that satisfy $f\leq_{\mathrm{lc}}g\leq_{\mathrm{lc}}h$, we present a pair of (reverse) triangle inequalities: if $\sum_iif_i=\sum_iig_i<\infty,$ then $$D(f|h)\geq D(f|g)+D(g|h)$$ and if $\sum_iig_i=\sum_iih_i<\infty,$ then $$D(h|f)\geq D(h|g)+D(g|f),$$ where $D(\cdot|\cdot)$ denotes the Kullback--Leibler divergence. These inequalities, interesting in themselves, are also applied to several problems, including maximum entropy characterizations of Poisson and binomial distributions and the best binomial approximation in relative entropy. We also present parallel results for continuous distributions and discuss the behavior of $\leq_{\mathrm{lc}}$ under convolution.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ216 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A Bit of Information Theory, and the Data Augmentation Algorithm Converges

The data augmentation (DA) algorithm is a simple and powerful tool in statistical computing. In this note basic information theory is used to prove a nontrivial convergence theorem for the DA algorithm

On a Multiplicative Algorithm for Computing Bayesian D-optimal Designs

We use the minorization-maximization principle (Lange, Hunter and Yang 2000) to establish the monotonicity of a multiplicative algorithm for computing Bayesian D-optimal designs. This proves a conjecture of Dette, Pepelyshev and Zhigljavsky (2008)

Some stochastic inequalities for weighted sums

We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let $Y_i$ be i.i.d. random variables on $\mathbf{R}_+$. Assuming that $\log Y_i$ has a log-concave density, we show that $\sum a_iY_i$ is stochastically smaller than $\sum b_iY_i$, if $(\log a_1,...,\log a_n)$ is majorized by $(\log b_1,...,\log b_n)$. On the other hand, assuming that $Y_i^p$ has a log-concave density for some $p>1$, we show that $\sum a_iY_i$ is stochastically larger than $\sum b_iY_i$, if $(a_1^q,...,a_n^q)$ is majorized by $(b_1^q,...,b_n^q)$, where $p^{-1}+q^{-1}=1$. These unify several stochastic ordering results for specific distributions. In particular, a conjecture of Hitczenko [Sankhy\={a} A 60 (1998) 171--175] on Weibull variables is proved. Potential applications in reliability and wireless communications are mentioned.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ302 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm