Population genetics and evolution have long supplied a stimulus to mathematical and statistical research. In this note a brief survey of current theory is given, with special reference to the work of P. A. P. Moran. Finally, an analysis of various biometrical questions is presented in the spirit of Moran's contributions to mathematical genetics.

Gene survival in a population which increases without density dependence is considered using a generalization of the Moran model for haploid individuals in which we have differential reproductive rates. It is shown under very mild conditions that ultimate fixation is certain. This provides some mathematical support for the phenomenon of reduced genetic variability in isolated populations established from founder groups.

A new Markov chain is introduced which can be used to describe the family relationships among n individuals drawn from a particular generation of a large haploid population. The properties of this process can be studied, simultaneously for all n, by coupling techniques. Recent results in neutral mutation theory are seen as consequences of the genealogy described by the chain.

We study the situation in which individuals occur in ‘families' or similar groups, individuals within a ‘family' being correlated with one another, as for example a biological population. In such a population, the number of individuals will usually vary from one family to another. We assume that a sample chosen from the population consists of whole families rather than unrelated individuals. A similar situation might occur in experimental design, if individuals occur in ‘blocks' (of varying sizes) within which they share a common environment. In the simplest case considered here two measurements are made on each individual, namely y, the character of interest, and x (not necessarily a random variable), some other character which is believed to influence y. We discuss how to estimate the regression of y on x, and the within and between family variance components for y (and hence the intrafamily correlation) when the effect of x is eliminated. Generalizations of this are briefly discussed.

It is shown that, for one model of genetic evolution, the times at which mutations at nucleotide sites become fixed in a population tend to cluster together rather than behave as a Poisson process.

This article discusses certain aspects of determining heritability estimates from the genetic analyses of data. First, a brief review is given of the major statistical techniques for fitting models to quantitative family data. Then, by reference to a general schematic framework, the basic models that have been proposed for such data will be described. Some new environmental models are also developed. Finally, by reference to some data sets, the interactive approach to data analysis is proposed, advocating the determination of adequacy of fit of all reasonable models to available data.

The question whether a convex closed curve can be approximated by ellipses having a large number of foci is considered. It is shown that the limiting, convex figure of multifocal ellipses may have only one single straight segment. This happens only in the case, when the foci tend partly to infinity and partly to points of the line through the straight segment. The approximations of certain ‘distance integrals' are treated; the characterization of approximability remains an open problem.

For each particle in an aggregate of point particles in the plane, the set of points having it as closest particle is a convex polygon, and the aggregate V of such Voronoi polygons tessellates the plane. The geometric and stochastic structure of a random Voronoi polygon relative to a homogeneous Poisson process is specified.

Similarly, those points of the plane possessing the same n nearest particles constitute a convex polygon cell in the generalized Voronoi tessellation 𝒱 (n = 2, 3, ·· ·). In fact, 𝒱 = 𝒱1, but to ease exposition n always takes the values 2, 3, ···. A key geometrical lemma elucidates the geometric structure of members of 𝒱n, showing it to be simpler in one important respect than that of members of 𝒱; in that, for each such N-gon of given ‘type', there is a uniquely determined set of N generating particles. The corresponding jacobian is given, and used to derive the basic ergodic structure of 𝒱n relative to a homogeneous Poisson process.

Unlike 𝒱 no 𝒱n contains any triangles. As n →∞, the vertices of the quadrangles of 𝒱n tend to circularity, so that the sums of their opposite interior angles tend to π.

A method used by electrical engineers to analyse polyphase alternating current systems suggests a generalisation to arbitrary plane polygons of a theorem on triangles nowadays known, for obscure reasons, as ‘Napoleon's Theorem': the centroids of equilateral triangles erected on the sides of an arbitrary triangle form the vertices of an equilateral triangle. The generalisation to other polygons uses a construction first studied by C.-A. Laisant in 1877; results of Jesse Douglas (1940) and the author (1941) are re-derived by means of the elementary algebra of finite-dimensional vector spaces over the field of complex numbers.

A survey is given of the attitude to and use of the concepts of deficient, perfect, and abundant number from the time of Nicomachus of Gerasa to that of David Anakht and Alcuin of York. Alcuin's ‘Letter to Dafnin', the source of an anecdote frequently mentioned in mathematical texts, is included as an appendix.

The statistics of deficient, perfect, and abundant numbers over the range 1–50000 are studied and presented graphically in several novel ways and compared with the work of Davenport and others on the distribution-law for the values of z = σ (n)/(2n).

Some queries are raised concerning observations which ancient writers might have been expected to make; for example, did they notice that about one half of all the even numbers are abundant?

There are described in the literature many spaces of what are variously described as generalized functions, distributions, or improper functions. This article introduces another. The new space is like that of M. J. Lighthill in containing the Fourier transform of every element and in having a particularly simple theory of trigonometric and Fourier series; also it is constructed in a somewhat similar way. The new space breaks away from the tradition of every element being, for some n, the nth derivative of an ordinary function, and, for example, the exponential function and its Fourier transform are in the space.

This paper examines the distribution of deaths in life tables for the population of England and Wales as a means of exploring the way in which the length of life is steadily advancing. To assess the possibility for future extension, some extreme assumptions are made about the reduction in mortality rates for certain causes. A distinction is made between ‘anticipated' deaths and ‘senescent' deaths, the latter group falling within bounds either side of a terminal peak in the distribution of deaths by age. For males, the extreme assumptions yield a peak at age 86 with some 85 per cent of deaths within the senescent area. For females, the peak is at about age 92 with 90 per cent of deaths in the senescent area.

A simple interaction model is proposed for describing variable meeting-rates in a population exposed to a contagious disease. In comparison with the usual assumption of a uniform meeting-rate it predicts, on average, a more rapid spread of infection initially and a slower spread finally; it also predicts that, for a given removal rate, an epidemic could develop among a smaller number of susceptibles.

The Cramér–Rao inequality for the variance of an unbiassed estimator is first recalled, and the value of the ‘ideal' equation of estimation existing if the minimum bound is reached illustrated by examples. When this estimation equation is not available, the more general inequality due to Kiefer is more relevant. The analogous estimation equation corresponding to the attaining of the more general minimum bound is again illustrated by examples; and the general theory, which is also available for more than one parameter, is indicated for any location and/or scaling problem.

A transformation is introduced to stabilize the variance of the largest value in a sample from a normal distribution. The transformed distribution is found to approximate closely to the limiting Gumbel form for all sample sizes. The practical implications of the result are discussed.

The asymptotic distribution of Hotelling's one-sample T2 in multivariate Edgeworth populations is expanded to terms of the first order. Comparison with published simulation results indicates that the result is quite useful, even in cases where the underlying population is not well represented by an Edgeworth expansion.

This paper considers functional dependence from a general standpoint. Some basic ideas are explored and the extent of functional dependence studied. These are illustrated by means of simple linear regression.

The standard linear regression model is analysed using a method called functional least squares which yields a family of estimators for the slope parameter indexed by a real variable t, |t| ≦ T. The choice t = 0 corresponds to ordinary least squares, non-zero values being appropriate if the error distribution is long-tailed, and it is argued that the approach is a natural extension of least squares methodology. It emerges that the asymptotic normal distribution of these estimators has a covariance matrix characterised by a scalar function of t, called the variance function, which is determined by the error distribution. Properties of this variance function suggest graphical criteria for detecting departures from normality.

A new approach to the minimization of polyhedral convex functions is applied to give a finite algorithm for the rank regression problem. Numerical results for the Daniel and Wood example are presented.