Normal variance-mean mixtures encompass a large family of useful
distributions such as the generalized hyperbolic distribution, which itself
includes the Student t, Laplace, hyperbolic, normal inverse Gaussian, and
variance gamma distributions as special cases. We study shape properties of
normal variance-mean mixtures, in both the univariate and multivariate cases,
and determine conditions for unimodality and log-concavity of the density
functions. This leads to a short proof of the unimodality of all generalized
hyperbolic densities.

One of two independent stochastic processes (arms) are to be selected at each
of n stages. The selection is sequential and depends on past observations as
well as the prior information. Observations from arm i are independent given a
distribution P_i, and, following Clayton and Berry (1985), P_i's have
independent Dirichlet process priors. The objective is to maximize the expected
future-discounted sum of the n observations. We study structural properties of
the bandit, in particular how the maximum expected payoff and the optimal
strategy vary with the Dirichlet process priors.

Comparison results are obtained for the inclusion probabilities in some
unequal probability sampling plans without replacement. For either successive
sampling or Hajek'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.

Improved EM strategies, based on the idea of efficient data augmentation
(Meng and van Dyk 1997, 1998), are presented for ML estimation of mixture
proportions. The resulting algorithms inherit the simplicity, ease of
implementation, and monotonic convergence properties of EM, but have
considerably improved speed. Because conventional EM tends to be slow when
there exists a large overlap between the mixture components, we can improve the
speed without sacrificing the simplicity or stability, if we can reformulate
the problem so as to reduce the amount of overlap.

One of the difficulties in calculating the capacity of certain Poisson
channels is that H(lambda), the entropy of the Poisson distribution with mean
lambda, is not available in a simple form. In this work we derive upper and
lower bounds for H(lambda) that are asymptotically tight and easy to compute.
The derivation of such bounds involves only simple probabilistic and analytic
tools. This complements the asymptotic expansions of Knessl (1998), Jacquet and
Szpankowski (1999), and Flajolet (1999).

A "cocktail algorithm" is proposed for numerical computation of (approximate)
D-optimal designs. This new algorithm combines and extends the multiplicative
algorithm of Silvey et al. (1978) and the vertex exchange method (VEM) of
Bohning (1986), and shares their simplicity and monotonic convergence
properties. Numerical examples show that the cocktail algorithm can lead to
dramatically improved speed, sometimes by orders of magnitude, relative to
either VEM or the multiplicative algorithm.

We investigate stochastic comparisons between exponential family
distributions and their mixtures with respect to the usual stochastic order,
the hazard rate order, the reversed hazard rate order, and the likelihood ratio
order. A general theorem based on the notion of relative log-concavity is shown
to unify various specific results for the Poisson, binomial, negative binomial,
and gamma distributions in recent literature.

We present an entropy comparison result concerning weighted sums of
independent and identically distributed random variables.

An inequality concerning ratios of gamma functions is proved. This answers a
question of Guo and Qi (2003).

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.