We consider planar first-passage percolation and show that the time constant can be bounded by multiples of the first and second tertiles of the weight distribution. As a consequence, we obtain a counter-example to a problem proposed by Alm and Deijfen (2015).

A selection of ten uni- and multivariate, nonparametric statistical methods thought to be useful for assessing trends in indicators estimated from trawl surveys is described. Nonparametric methods make minimal assumptions about the data and can therefore be suitable when parametric modelling methods, as typically preferred by fisheries scientists, are not suitable. The various methods described are sensitive to a variety of specific features of a trend thereby allowing many different forms of variation over time to be identified, e.g. changes in level, sloping straight or curved lines, and differing cyclicities. In contrast, model-based investigations can be constrained in this regard if the single model which is adopted and fitted to the trend is a simplification of its form.

The Lewis-Thompson (L-T) approach to ordering distributions in terms of dispersion is compared to the dispersion ordering obtained from majorisation. For given conditions, L-T ordering is shown to be stronger than majorisation ordering. Dilation is considered in relation to majorisation ordering.

Let {Zn} be a finite mean supercritical Bienaymé– Galton–Watson process. It is known that there exist norming constants {Cn} such that {Zn/Cn} converges almost surely to a limit W. Also there is a whole literature concerning properties of {Cn} and W. We attempt a new approach to the limit theory of {Zn} by relating it to the theory of sums of independent and identically distributed random variables.

As an approach to modelling the ‘matching’ of optical receptors in animals to the objects they are designed to see, we study the problem of locating regions of high intensity in a point process on the real line, using the counts of points in a movable interval of fixed length. We define performance measures analogous to statistical size and power for this procedure and, for points forming a renewal process, give conditions on the quantiles of the convolutions of the interpoint distribution which ensure that the optimal length for the ‘detector’ is close to that of the ‘object’ to be detected. We show that these conditions are satisfied for a Poisson process. Similar conditions ensure that the optimal length is close to zero, and we give a class of distributions satisfying these conditions. Finally we show that the results can be extended to simple two-dimensional models.

We show, in the context of a neurobiological problem, that ζ a(α)−ζ a(β) increases with a, while ζ a (α)/ζ a (β) decreases with a, where 0 < β < α < 1 and ζ a (α) is the α th quantile of either the gamma distribution with shape parameter a, or a scaled F distribution with parameters 2a and 2m for any m.