Consider situations where the treatment may cause an initial effect and may also cause a long-range effect. We want to evaluate the treatment, or to compare two treatments, when the effect of treatment may result from the two distinct mechanisms, M1 and M2. We may wish to evaluate M1 and M2 separately, but we may also want to evaluate their combined effect M3. Examples are given and the general results are applied to the special case arising in weather modification studies and elsewhere: the possible effects are multiplicative and the distribution of non-zero variables is gamma with at most the scale parameter affected by treatment. An example demonstrates that the two components may be too weak to be judged significant while their sum is large and significant. The locally optimum C(α) test is used.

There is a brief discussion of the power function of the tests. The asymptotic power agrees well, in general, by the results of the Monte Carlo simulation for the test Z3 of the combined effect. If the zero values are discarded and then Z2 employed, there is large bias in the power. The bias is more pronounced if the Wilcoxon, Mann–Whitney test is employed. Notice that the two effects under study may be acting in the same direction or they may be in opposition.

A survey is made of the mathematical properties of, and the arithmetic relationships between, various distributions on the circle and the sphere. The Brownian motion and angular Gaussian distributions are shown in computer-drawn graphs to bracket the von Mises–Fisher distributions.

This paper derives upper and lower bounds to the distribution functions of nearest-neighbour and minimum nearest-neighbour distances between N points generated by a hard-core model on the surface of a sphere. The use of these bounds in statistical inference is discussed.

Data classified into several groups usually depend on structural and accessory parameters, the structural parameter θ being a constant for all the data, while the accessory parameters ϕ i vary between groups.

For the structural parameter, inferences that are free of the accessory parameters can be made when the ϕ i admit sufficient statistics or surrogates Si, by conditioning on the Si or by constructing statistics independent of the Si.

When the Si are functions of the structural parameter θ, θ is said to be endomorphic. Conditional likelihood methods are not unequivocal for endomorphic parameters; instead, inference is based on statistics independent of the Si, and so free of the ϕ i, derived from conditional distribution functions.

With endomorphic parameters, since the surrogates and the statistics independent of them are functions of θ, tests of independence of these sets of statistics provide a means of making inferences about θ.

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.

Null-recurrent random walks with symmetric step distribution are mapped to give null-recurrent Markov chains on the interval (0, 1). These chains are used to construct stationary Wold processes (point processes with Markov-dependent intervals) with infinite intensity, which processes are simple (i.e., almost surely orderly) but not (analytically) orderly. As examples of point processes with infinite intensity, they are constructed without any intermediate step that relies on using point processes of finite intensity.

A direct proof of the expression for the limit probability generating function (p.g.f.) of the sum of Markov Bernoulli random variables is outlined. This depends on the larger eigenvalue of the transition probability matrix of their Markov chain.

This paper discusses the asymptotic behaviour, and in particular the critical singularities, of the generating functions for Pólya walks with a given fraction of visits to a hyperplane when the walk is or is not constrained to be on one side of that hyperplane. The three-dimensional case has applications in physics and chemistry.

Over a finite interval, a Gaussian stationary process can be approximated by a finite trigonometric sum, and the error introduced by the approximation can be exactly bounded, as far as the distribution of the upper tail of the maximum is concerned. A simple case is exhibited, where a narrow band process is well approximated by means of a two-term trigonometric representation.

In previous papers (1976), (1977a), (1979) limit theorems were obtained for the classical Ising model, and these provided the basis for asymptotic inference. The present paper extends these results to more general Ising models.

In two and more dimensions, likelihood inference for the thermodynamic parameters (i.e. the interaction energies) is effectively impossible. The problem is that the error in locating critical and/or confidence regions is as large as their diameters. To remedy this requires more accurate characterizations of the partition functions, but these seem unlikely to be forthcoming. Besag's coding estimators for these parameters are inverse hyperbolic tangents of the roots of simultaneous polynomial equations and hence avoid such location errors. However, little is yet known about their sampling characteristics. Finally, likelihood inference for lattice averages (an alternative parametrization) is straightforward from the limit theorems.

The theory of weak convergence has developed into an extensive and useful, but technical, subject. One of its most important applications is in the study of empirical distribution functions: the explication of the asymptotic behavior of the Kolmogorov goodness-of-fit statistic is one of its greatest successes. In this article a simple method for understanding this aspect of the subject is sketched. The starting point is Doob's heuristic approach to the Kolmogorov-Smirnov theorems, and the rigorous justification of that approach offered by Donsker. The ideas can be carried over to other applications of weak convergence theory.

The concept of entropy in models is discussed with particular reference to the work of P.A.P. Moran. For a vector-valued Markov chain {Xk} whose states are relative-frequency (proportion) tables corresponding to a physical mixing model of a number N of particles over n urns, the definition of entropy may be based on the usual information-theoretic concept applied to the probability distribution given by the expectation . The model is used for a brief probabilistic assessment of the relationship between Boltzmann's Η-Theorem, the Ehrenfest urn model, and Poincaré's considerations on the mixing of liquids and card shuffling, centred on the property of an ultimately uniform distribution of a single particle. It is then generalized to the situation where the total number of particles fluctuates over time, and martingale results are used to establish convergence for .

Suppose that it is desired to determine the frequency, known to lie in a range 0 < σ ≦ ω ≦ ΩT <∞, of a periodic effect modulating the rate of occurrence of a Poisson point process. It is shown that if ΩT increases sufficiently slowly with T, the length of the observation period, then the frequency corresponding to the maximum of the Bartlett periodogram over this range provides a consistent estimate of the unknown frequency. The asymptotic covariance matrix is found for the maximum likelihood estimates of the parameters in a cyclic Poisson model, and the close analogue with frequency estimation for Gaussian processes is emphasized. It is pointed out, however, that in the point-process case periodogram estimates will be efficient only for very special models.

A general form of Green's theorem is used to derive relations between expected costs for an optimisation problem and the distribution of state variable. A characterisation of the optimal stopping set and an alternative proof of the Howard improvement lemma emerge as non-trivial consequences.

The problem initially considered is that of testing whether ρ = 0 in a model y(n) = x(n)+ η (n), x(n) = ρx(n – 1) + ξ (n) where only y(n) is observed and η (n), ξ (n) are white noise. This is equivalent to distinguishing between an ARMA (1, 1) model and white noise. The asymptotic distribution of the likelihood ratio criterion is derived. This is shown to be of an unusual form. This result is then used to discuss the asymptotic properties of Akaike's procedure for estimating (p, q) in an ARMA (p, q) model. If p0, q0 are the true values and p0 < P, q0< Q, when P, Q are the maximum values considered, then it is shown that, in a certain asymptotic sense, the procedure is sure to overestimate p0, q0. However, the asymptotic situation may be very far from that relevant in a practical case. The relevance of overestimation is briefly discussed.

Some inadequacies of both the traditional (exponential smoothing) and Box-Jenkins approaches to time series forecasting of economic data are investigated. An approach is suggested which integrates these two methodologies. It is based on smoothing the data using straight line segments instead of differencing to obtain stationarity, and forecasting using an autoregressive-moving-average model for the residuals from the most recent linear segment. The efficiency of this approach is calculated theoretically using a series comprising integrated white noise.