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Summary Fisher has presented various applications of the repeated integral of the normal distribution function. A new application has appeared in shape distributions. This work has led to the search for an explicit expression for the repeated normal integral function in place of an infinite series expansion for this function first given by Fisher. We provide an explicit expression for this function with a finite number of terms.

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

In this paper we study the distribution associated with paired size-and-shape for two sets of correlated points having an underlying Gaussian distribution in the plane. This extends work previously done just for paired shape. Special cases are studied, and various properties described. In addition we give further results for the paired shape density.

The paper considers the bias of Bookstein's mean estimator for shape under the isotropic normal model. This work depends on certain distributional properties of shape variables. An alternative unbiased modified estimator is proposed and its performance is compared with various estimators, including Procrustes and the exact maximum likelihood estimator, in a simulation study.

The Ising model has become more prominent in spatial statistics since its applications in image analysis, as pioneered by Besag. This paper describes three multilevel generalizations of the Ising model, including the general spin Ising model. We compare and contrast these three generalizations. In all three cases, the normalizing constant is intractable but, using the developments made by physicists, we give adequate approximations for the general spin model, together with complete expressions for the binary Ising model. We show that these approximations allow inference for the general spin model. An application to texture analysis is also given.

In this paper we investigate the exact shape distribution for general Gaussian labelled point configurations in two dimensions. The shape density is written in a closed form, in terms of Kendall's or Bookstein's shape variables. The distribution simplifies considerably in certain cases, including the complex normal, isotropic, circular Markov and equal means cases. Various asymptotic properties of the distribution are investigated, including a large variation distribution and the normal approximation for small variations. The triangle case is considered in particular detail, and we compare the density with simulated densities for some examples. Finally, we consider inference problems, with an application in biology.

SUMMARY Rao's score statistic is a standard tool for carrying out hypothesis tests. A unified approach is given which encompasses many common tests. When a suitable group structure is present, it is shown how the derivation of the test statistic can be simplified. An omnibus test of multivariate normality is derived as an example of group invariance. As another application of Rao score tests, a general method is given for constructing tests of independence.

Time series the asymptotic distribution of auto-correlation co-efficients directional data analysis on Watson's anova for directions compositional and shape data analysis the triangle in statistics new directions for shape analysis technical problems in inference tests of fit for logistic models spatial statistics how to charge for boundaries in a Pixel image statistics and genetics case studies on issues of public policy.

The paper obtains the exact distribution of Bookstein's shape variables under his plausible model for landmark data. We consider its properties including invariances, marginal distributions and the relationship with Kendall's uniform measure. Particular cases for triangles and quadrilaterals are highlighted. A normal approximation to the distribution is obtained, extending Bookstein's result for three landmarks. The adequacy of these approximations is also studied.

Summary. Consider a sequence of, say, 10 to 20 vector observations in threedimensional space. It is suspected that a few subsets of consecutive observations are made up of collinear points. The purpose of this paper is to construct a statistically based algorithm to find such linear segments and to assess their accuracy. A similar assessment is made for coplanar sets of points. This algorithm is applied here to palaeomagnetic data and is claimed to be superior to previous methods of palaeomagnetic analysis in terms of completeness and balance of analysis, treatment of measurement errors and other sources of scatter, criteria for identification of linear and planar sets of points, and statistical rigour. Stability spectra, with statistically based confidence limits, are obtained as a by-product. 1 Palaeomagnetic analysis: background to the geometrical problem The natural remanent magnetization (NRM) in rocks is known in general to have originated in one or more relatively short time intervals during their formation and subsequent history, i.e. during rock-forming or metamorphic events during which NRM is frozen-in (stabilized) by falling temperature, grain growth etc. The NRM acquired during each such event is a single vector magnetization parallel to the then-prevailing geomagnetic field and is called a -component of NRM’. Rocks which have suffered more than one event of this kind are therefore liable to carry ‘multi-component NRM’. In the laboratory these components can be identified by demagnetizing the rock step-by-step, usually by thermal, alternating field or chemical demagnetization (which involves the analysis of the distribution of blocking rzmperature, coercivity and chemical leaching respectively). Resistance to these treatments is called ‘stability’ of remanence, and the range of treatment over which a particular component is removed is called the stability range of the component. The various discrete

The paper considers the dam problem with Markovian inputs, with special reference to the serial correlation coefficient of the input process. An input model is proposed which by giving particular values to the parameters makes the stationary distribution of the inputs one of the standard discrete distributions. The probabilities of first emptiness before overflow are first obtained by using the Markovian analogue of Wald's Identity. From these, the stationary distributions of the dam content are obtained by a duality argument. Both, finite and infinite dams are considered. MARKOVIAN INPUT OF LINEAR REGRESSIVE KIND; MARKOVIAN ANALOGUE OF WALD'S IDENTITY; PROBABILITIES OF EMPTINESS; STATIONARY DISTRIBUTIONS OF DAM CONTENT; DEPENDENCE ON SERIAL CORRELATION COEFFICIENT

Let X and Y be the minimum and maximum observations in a random sample of size n drawn from a continuous population having the probability density function (p.d.f.) f (x) and distribution function F(x). Let the sample range be R(= Y--X) . For convenience, we shall assume tha t x is limited by a

The von Mises-Fisher distribution is the most important distribution in directional data analysis. We derive the sampling distributions of the sample resultant length, the sample mean direction and the component lengths. For the multi-sample case, the conditional distribution of the individual sample resultant lengths given the combined sample resultant length is derived. These results depend heavily on the corresponding distributions for the isotropic random walk on hypersphere. Using these results we investigate some optimum properties of various important tests. Most of these tests were formulated intuitively by Watson and Williams (1956). Mardia (1972) in his book concentrated on the optimum properties of the circular and spherical cases, and this paper extends and unifies some of the parametric work.

In 1918 von Mises discovered the circular probability distribution which now bears his name by studying the analogue on the circle of the Gaussian maximum likelihood characterization of the normal distribution on the line. Teicher [6] gave a specific condition on the density function for the Gaussian characterization to hold (see also Kagan, et al. [2], p, 411). It is shown in this paper that the same condition is sufficient for the von Mises characterization to be valid. The proof is based on Teicher’s approach, but the periodic nature of circular densities makes substantial modifications necessary. An extension to the hyperspherical case is also given.

The paper sets out to investigate a statistical problem in the geophysical field of plate tectonics which amounts to obtaining a small circle of best fit in some sense. We define a small circle of best fit. Using the maximum entropy principle, an appropriate distribution is constructed, leading to the setting up of a confidence zone for this small circle. Finally, detailed analyses for a set of data in the field of plate tectonics and for a set of data concerning secular variation of the palaeomagnetic field are given.

WHY multivariate analysis should be so popular is uncertain, but the choice of conference was clear. The interest in multivariate analysis had been reflected earlier, in 1970, with the formation of the Multivariate Study Group, and it was decided that the General Applications Section and Multivariate Study Group should jointly organize the conference. It is not easy to arrange a programme that interests a heterogeneous audience ranging from the knowledgeable to the not-so-knowledgeable and the activities of the M.V.S.G. had to some extent pre-empted certain topics, at least for London-based statisticians. It was thought that a series of papers on some of the main topics of current interest should be presented, allowing plenty of discussion time and, hopefully, presenting several case studies. The final programme, to some extent, consisted of a series of review papers that we hoped would be useful both as revision and as introductions to parts of a subject that may be little known to many statisticians. We think that there was room for a few more case studies. The word multivariate was interpreted fairly broadly so that, for example, regression problems and their associated linear models for contingency tables could be included. This programme was evidently popular as there were over 300 applications for a maximum of 250 places available at Hull University's Lawns Conference Centre.