In this paper the concavity property for the distribution of a geometric random sum (geometric compound) X, + · ·· + XN is established under the assumption that Xi are i.i.d. and have a DFR distribution. From this and the fact that the actual waiting time in GI/G/1 queues can be written as a geometric random sum, the concavity of the waiting-time distribution in GI/G/1 queues with a DFR service-time distribution is derived. The DFR property of the idle-period distribution in specialized GI/G/1 queues is also mentioned.

We consider the action of independent and identically distributed n × n circulants S1, S2, … on V = [v1, …, vn] whose columns are the positions of n points in ℝd. The positions of the n points after m transformations are the columns of W(m) = VS1 … Sm. We describe, in several ways, the shape of the configuration of the points W(m) as m →∞. When n = 3, 4 and d = 2, a special discussion in terms of Moebius transformations is given.

A risk-sensitive certainty equivalence principle is deduced, expressed in Theorem 1, for a model with linear dynamics and observation rules, Gaussian noise and an exponential-quadratic criterion of the form (2). The senses in which one is now to understand certainty equivalence and the separation principle are discussed.

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

The scope of statistical packages is largely defined by the data structures supported. It is claimed that none of the standard structures used in major packages is wholly suitable for time series data and this is the major reason why time series has not benefited from modern developments in statistical computing.

A method is given for calculating the probability that n terms of a Gaussian first-order autoregressive series lie between any specified limits where n may be as large as 20 or more.

An algorithm for exponential fitting is presented which exploits the separable regression structure and a reparametrization. The algorithm has proved very satisfactory, and theoretical reasons for this are developed.