bluuebunny/crossref_metadata_2025 huggingface dataset:
The Two-Pack Matching Problem
Authors: Joseph A. W. , Bizley M. T. L.
Published: December 2018

The Two-pack Matching Problem is described. The number of derangements is obtained by applying an adaptation of the Laguerre polynomial considered as an operator and this leads to explicit formulae and generating functions for the number of r-hits. Some useful properties of the adapted Laguerre polynomials are given from which recurrence formulae are developed for the number of derangements both generally and also when the pack consists of n different sets of p like cards. It is shown how to build up a hit generating function stage by stage by a simple numerical process. Tables of numbers of derangements and r-hits are given in an Appendix.
Models in the Analysis of Variance
Authors: Plackett R. L.
Published: December 2018

A survey is given of recent developments in the analysis of variance of factorial experiments, with particular reference to randomization models. The effect of the assumptions on expected mean squares is outlined, and the problems which arise when estimating components of variation or applying tests of significance are discussed.
Some Notes on Pistimetric Inference
Authors: Roy A. D.
Published: December 2018

The Pistimetric method, in which the probability distribution of parameters is determined by means of a pivotal quantity as soon as the number of observations permit, and posterior probability distributions are subsequently deduced by Bayes’s theorem, is here applied in outline to a variety of situations. These include cases where the variate is discrete, where it is continuous, and when events occur in continuous time.
The Moment Generating Function of the Truncated Multi-Normal Distribution
Authors: Tallis G. M.
Published: December 2018

In this paper the moment generating function (m.g.f.) of the truncated n-dimensional normal distribution is obtained. From the m.g.f., formulae for E(Xi) and E(Xi Xj) are derived, and are used to investigate certain special cases. Some applications of these results to statistical genetics are also discussed.
Testing for a Jump in the Spectral Function
Authors: Hannan E. J.
Published: December 2018

The problem of testing for a jump in the spectral function of a stationary time series is considered. The test based on the greatest periodogram ordinate divided by an estimate of the spectral density is discussed and a modification is suggested which consists essentially of taking out regression on the harmonic with frequency 2πj/n, before computing the estimate of the spectral density at that frequency. The regression can be approximately effected by using the corresponding periodogram ordinate.
A Note on Maximum-Likelihood in the Case of Dependent Random Variables
Authors: Silvey S. D.
Published: December 2018

The validity of certain statistical procedures depends on the “classical” properties of the method of maximum-likelihood, viz., its consistency and the fact that it leads to an estimator whose asymptotic variance can be derived easily from the likelihood function. These properties are well established for the case where successive observations are independent. Here it is shown that their extension to the dependent case involves suitable behaviour of two semi-martingales and a martingale, and general conditions are given which ensure this behaviour. While interest is centred on the theoretical rather than on the practical side of the problem, these conditions may have some practical value.
Similarities between Likelihoods and Associated Distributions a Posteriori
Authors: Sprott D. A.
Published: December 2018

Some methods of constructing distributions a posteriori are based on rules of mathematical manipulation that ignore the logical role played by the likelihood. The effect of such methods has been to alter the relationship between variates as they occurred in the likelihood, and so change what may be called their relative weight. Thus, although in a sense no information has been lost, there certainly has occurred a distortion. The concept of relative weight is examined in the case of one parameter, and is used to investigate, in a particular example, what happens logically to samples that are combined by Bayes's theorem. The multiparameter case is also examined in an effort to set up methods of constructing such distributions a posteriori that will preserve the essential form of the likelihood, and so perhaps qualify as genuine fiducial distributions. Examples are given from the literature in which this is not the case.
Efficient Estimates and Optimum Inference Procedures in Large Samples
Authors: Rao C. Radhakrishna
Published: December 2018

The concept of efficiency in estimation is linked with closeness of approximation to the derivative of log likelihood, which plays an important role in statistical inference in large samples. Various orders of efficiency are defined depending on degrees of closeness, and properties of estimates satisfying these criteria are studied. Such measures of efficiency appear to be more appropriate than the one related to asymptotic variance of an estimate for judging the performance of an estimate, when used as a substitute for the whole sample in drawing inference about unknown parameters. It is found that, under some conditions, the maximum likelihood estimate has some optimum properties which distinguish it from all other large sample estimates.
Time-Dependent Solution of the Head-Of-The-Line Priority Queue
Authors: Jaiswal N. K.
Published: December 2018

The method of supplementary variables has been used to obtain the Laplace transform of the time-dependent probabilities in a head-of-the-line priority queue characterized by Poisson arrivals and general service-time distributions. An explicit solution has, however, been obtained under exponential service-time distributions with equal mean rates. The queue length probabilities, under steady-state conditions, have been evaluated and are found to be different from those obtained by Miller, thus showing that for complex queues, the asymptotic probability distribution obtained by considering the evolution of the queueing process in continuous time need not be the same as the one obtained by the imbedded Markov chain technique.
Fiducial Limits for a Variance Component
Authors: Healy M. J. R.
Published: December 2018

The exact fiducial distribution is obtained for a variance component, and compared with a previous approximation.
The Simplex-Centroid Design for Experiments with Mixtures
Authors: Scheffé Henry
Published: December 2018

A new design for experiments with mixtures of q components comprises the 2q − 1 points representing the mixtures consisting of all possible subsets of the q components, present in equal proportions. A suitable regression equation is proposed and fitted. Designs and regression equations permitting the introduction of n “process variables” in addition to the q “mixture variables” are considered, and fractional replication of the designs in the case where the process variables are all at two levels.
On Formulae for Confidence Points Based on Integrals of Weighted Likelihoods
Authors: Welch B. L. , Peers H. W.
Published: December 2018

Lindley’s problem, relating to the conditions under which there is formal mathematical equivalence between Bayesian solutions and confidence theory solutions, is discussed. A result concerning location parameters is extended. The corresponding asymptotic theory is developed and a modification of the problem is also discussed.
On the Monotonicity Property of the Three Main Tests for Multivariate Analysis of Variance
Authors: Srivastava J. N.
Published: December 2018

This paper presents a unified and relatively simple proof of the monotonicity property of the three well-known tests for the multivariate analysis of variance (MANOVA), viz., (i) Hotelling-Lawley trace criterion, (ii) Wilks's likelihood-ratio criterion, and (iii) Roy's largest root criterion. The last one has already been considered by Roy and Mikhail (1961). Their proof, however, has a gap at one place, which has been bridged in this paper.
Characterization Theorems for Some Univariate Probability Distributions
Authors: Patil G. P. , Seshadri V.
Published: December 2018

Suppose that one is dealing with a bivariate distribution of random variables X and Y where only the conditional distribution of X given X + Y is known. What can be said of the marginal distributions of X and Y? This paper attempts to answer this question for situations involving certain univariate distributions known in the statistical literature.
On Certain Properties of the Exponential-Type Families
Authors: Patil G. P. , Shorrock Richard
Published: December 2018

Certain structural properties of the exponential-type families are studied under three headings: (1) mean value function and the exponential-type families, (2) a characterization of the Gamma family and (3) the quality of the first two Bhattacharya bounds and the exponential-type families.
On a Property of the Random Walks Describing Simple Queues and Dams
Authors: Loynes R. M.
Published: December 2018

A so-called duality principle, which has been given by previous authors for simple queues and dams, is here shown to be valid in more general circumstances. The conditions under which the result, relating the stationary distribution of one random walk with absorption probabilities in another, is applicable are investigated.
On the Estimation of the Intensity Function of a Stationary Point Process
Authors: Cox D. R.
Published: December 2018

The intensity function of a stationary point process gives as a function of x the conditional probability, given an event at time t, of an event in the interval (t + x, t + x + Δx). A very nearly unbiased estimate of the grouped intensity function is constructed. its sampling properties are investigated when the data are generated by a Poisson process.
A General Class of Coefficients of Divergence of One Distribution from Another
Authors: Ali S. M. , Silvey S. D.
Published: December 2018

Let p1 and p2 be two probability measures on the same space and let φ be the generalized Radon-Nikodym derivative of p2 with respect to p1. If C is a continuous convex function of a real variable such that the p1-expectation (generalized as in Section 3) of C(φ) provides a reasonable coefficient of the p1-dispersion of φ, then this expectation has basic properties which it is natural to demand of a coefficient of divergence of p2 from p1. A general class of coefficients of divergence is generated in this way and it is shown that various available measures of divergence, distance, discriminatory information, etc., are members of this class.
Characterization Theorems for Some Univariate Probability Distributions
Authors: Menon M. V.
Published: December 2018

This paper is an extension of the work of Patil and Seshadri (1964), and deals with the problem of characterizing the distribution of two independent random variables X and Y when the conditional distribution of X given X + Y is known.
Design Relations for Non-Stationary Processes
Authors: Priestley M. B.
Published: December 2018

Further developments in the theory of evolutionary spectra are discussed, and some asymptotic results are derived in the sampling theory of spectral estimates. Various forms of “design” criteria for spectral estimation are constructed in terms of resolvability conditions, and it is shown that by introducing the concepts of “frequency-domain” and “time-domain” bandwidths, estimates may be obtained which satisfy the required conditions.
Exponential Smoothing for Multivariate Time Series
Authors: Jones Richard H.
Published: December 2018

A method is presented for estimating the optimum weight matrix for the exponential smoothing and prediction of multivariate time series. A recursive version of the estimation equations is given. A variation allows the estimation to forget the remote past in order to follow a process in which the structure varies slowly with time. It is pointed out that the method generalizes to non-linear processes when the non-linear structure is known. The practicality of the method is demonstrated by a numerical simulation study.
Some Applications of Probability Generating Functionals to the Study of Input-Output Streams
Authors: Vere-Jones D.
Published: December 2018

Probability generating functionals are used to obtain simple proofs of Mirasol's theorem on the output to an M/G/∞ queue, the limit theorem for the superposition of point processes, and a limit theorem for the output from a sequence of G/G/∞ queues.
A Test for Non-Stationarity of Time-Series
Authors: Priestley M. B. , Rao T. Subba
Published: December 2018

We consider the problem of testing a given time-series for stationarity. The approach is based on evolutionary spectral analysis, and the proposed method consists essentially in testing the “homogeneity” of a set of evolutionary spectra evaluated at different instants of time.

Using a logarithmic transformation, we show that the mechanics of the test are formally equivalent to a two-factor analysis of variance procedure when the residual variance is known, a priori.

In addition to testing stationarity, the analysis provides also a method for testing whether the observed series fits a “uniformly modulated” model, and a test for “randomness” (constancy of spectra).
Exact Bayesian Analysis of a 2 × 2 Contingency Table, and Fisher's “Exact” Significance Test
Authors: Altham Patricia M. E.
Published: December 2018

A relationship is derived between the posterior probability of negative association of rows and columns of a 2 × 2 contingency table and Fisher's “exact” probability, as given in existing tables for testing the hypothesis of no association of rows and columns. The result for the 2 × 2 table is generalized to provide the posterior probability that one discrete-valued random variable is stochastically larger than another.
On Distributions and Their Transformation to Normality
Authors: Draper N. R. , Cox D. R.
Published: December 2018

Further consideration is given to a family of power transformations discussed by Box and Cox (1964). It is shown that this family of transformations can be useful even in situations where no power transformation can produce normality exactly. The precision of estimation of the transformation parameter is also discussed.
Inequalities in the Theory of Queues
Authors: Kingman J. F. C.
Published: December 2018

Some simple inequalities are established for the stationary waiting time distribution in the queue GI/G/k, both for the single-server case k = 1 and for general values of k.
Application of Likelihood Methods to Models Involving Large Numbers of Parameters
Authors: Kalbfleisch John D. , Sprott D. A.
Published: December 2018

Likelihood methods of dealing with some multiparameter problems are introduced and exemplified. Specifically, methods of eliminating nuisance parameters from the likelihood function so that inferences can be made about the parameters of interest are considered. In this regard integrated likelihoods, maximum relative likelihoods, conditional likelihoods, marginal likelihoods and second-order likelihoods are introduced and their uses illustrated in examples. Marginal and conditional likelihoods are dependent upon factorings of the likelihood function. They are applied to the linear functional relationship and to related models and are found to give intuitively appealing results.

These methods indicate that in many situations commonly encountered objective methods of eliminating unwanted parameters from the likelihood function can be adopted. This gives an alternative method of interpreting multiparameter likelihoods to that offered by the Bayesian approach.
Further Consideration of the Distribution of the Multiple Correlation Coefficient
Authors: Gurland John , Milton Roy
Published: December 2018

A general class of series expansions for the non-null distribution of the multiple correlation coefficient is presented which contains, as special cases, the three expansions obtained by one of the authors (Gurland, 1968). The rate of convergence is investigated, and the closeness of an approximation to the true distribution is indicated.
Bivariate Exponential Distributions in Reliability Theory
Authors: Downton F.
Published: December 2018

A simple failure model is used to derive a bivariate exponential distribution. This distribution, although well known in the literature, does not appear to have been considered in a reliability context. its properties are considered and in particular explicit expressions are obtained for the distributions of the larger and of the smaller of a pair of correlated exponential observations. The results are compared with those obtained by Marshall and Olkin (1967a, b) for a different bivariate exponential model motivated by reliability theory.
Some Results on the Sampling Distribution of the Multiple Correlation Coefficient
Authors: Lee Yoong-Sin
Published: December 2018

Recurrence formulae for the density function and the probability integral of the multiple correlation coefficient from a normal sample are obtained. When the number of independent variates is odd these formulae are related to the distribution of the simple correlation. Asymptotic expansions in terms of the non-central beta and of the non-central χ2 distributions are derived. Methods of approximation by the central and the non-central F distributions, as well as by the normal distribution, are proposed. Numerical investigations of the accuracies of these approximations are carried out.
Point Estimation of Forces of Transition in Demographic Models
Authors: Hoem Jan M.
Published: December 2018

Occurrence/exposure rates are developed as estimators for the forces of transition of a homogeneous, time-continuous Markov process with at most countable state space in the case where the number of such forces is finite. Conditions for consistency and asymptotic normality of the estimators are stated.
Formal Expressions Which Can be Applied to Cusum Charts
Authors: Kemp K. W.
Published: December 2018

The paper describes a number of formal expressions which can be used to determine properties of cumulative sum schemes. Formulae given only depend upon very general assumptions with regard to the distribution of the cumulated statistic. Numerical as well as algebraic procedures are described which permit calculation of the average sample run length to a specified degree of accuracy for almost any statistic likely to arise in practice. Methods for generating the sample run length distribution are also given.
Components of Cramér–Von Mises Statistics. I
Authors: Durbin J. , Knott M.
Published: December 2018

Let Fn(x) be the sample distribution function derived from a sample of independent uniform (0, 1) variables. The paper is mainly concerned with the orthogonal representation of the Cramér-von Mises statistic Wn2 in the form ∑j=1∞(jπ)−2znj2 where the znj are the principal components of n{Fn(x)−x}. It is shown that the znj are identically distributed for each n and their significance points are tabulated. Their use for testing goodness of fit is discussed and their asymptotic powers are compared with those of Wn2, Anderson and Darling's statistic An2 and Watson's Un2 against shifts of mean and variance in a normal distribution. The asymptotic significance points of the residual statistic Wn2−∑j=1p(jπ)−2znj2 are also given for various p. It is shown that the components analogous to znj for An2 are the Legendre polynomial components introduced by Neyman as the basis for his “smooth” test of goodness of fit. The relationship of the components to a Fourier series analysis of Fn(x) –x is discussed. An alternative set of components derived from Pyke's modification of the sample distribution function is considered. Tests based on the components znj are applied to data on coalmining disasters.
Resolution Iv Designs of the 2M × 3 Series
Authors: Anderson D. A. , Srivastava J. N.
Published: December 2018

It is shown that a resolution IV 2m × 3 factorial design without blocks permits the estimation of not only the main effects A1, …, Am, B and B2 (which are assumed estimable by definition) but also of (i) the interactions Ai B, and Ai B2 (i = 1, …, m), and (ii) at least one linear combination of the general mean μ and the interactions Ai Ai (i < j; i, j = 1, …, m). Some methods of construction of 2m × 3 designs of resolution IV are discussed and “good” designs of this series with N = 4N* are presented for each m in the range 3 ⩽ m ⩽ 15, and N ⩽ 80. These designs may be conducted in blocks of size 4 or any multiple of 4.
Generalized Least Squares and Maximum Likelihood Estimation of Non-Linear Functional Relationships
Authors: Dolby G. R.
Published: December 2018

A non-linear functional relationship between two mathematical variables is postulated. Departures of observed random variates from the mathematical variable values are assumed to have zero mean and general but known covariance matrix. Assuming normality we obtain by maximum likelihood an expression for the asymptotic covariance matrix of the structural parameter estimates. The expression involves a certain matrix Ψ. A generalized least squares method is proposed which involves minimizing a quadratic form with the same matrix Ψ. In the linear case, it is shown that the methods of generalized least squares, maximum likelihood and maximum relative likelihood are equivalent. The least squares procedure is identified with the intuitive approach of Sprent (1966).
Numerical Results on Approximate Confidence Limits for the Odds Ratio
Authors: Gart John J. , Thomas Donald G.
Published: December 2018

Consider the comparison of two binomial variates with parameters p1 and p2. The parameter of non-centrality for the Fisher–Irwin exact test is the odds ratio, ψ = p1q2/p2q1, where qi = 1 – pi, i = 1, 2. Exact confidence limits for ψ may be calculated (Fisher, 1935, 1962; Cornfield, 1956) and several authors have suggested approximate methods. The various approximate methods are reviewed and, from both theoretical and numerical results, we find that Cornfield's approximate method and Method II of Gart (1962) yield the better approximations under various conditions. An F-test alternative to the exact test of 2 × 2 tables (Gart, 1962) is found to be accurate in small numbers.
Combining Possibly Related Estimation Problems
Authors: Efron B. , Morris C.
Published: December 2018

We have two sets of parameters we wish to estimate, and wonder whether the James-Stein estimator should be applied separately to the two sets or once to the combined problem. We show that there is a class of compromise estimators, Bayesian in nature, which will usually be preferred to either alternative.

“The difficulty here is to know what problems are to be combined together— why should not all our estimation problems be lumped together into one grand melée?”

George Barnard commenting on the James–Stein estimator, 1962.
Assessing Uncertainty
Authors: Tversky Amos
Published: December 2018

Intuitive judgments of probability are based on a limited number of heuristics that are usually effective but sometimes lead to severe and systematic errors. Research shows, for example, that people judge the probability of a hypothesis by the degree to which it represents the evidence, with little or no regard for its prior probability. Other heuristics lead to an overestimation of the probabilities of highly available or salient events, and to overconfidence in the assessment of subjective probability distributions. These biases are not readily corrected, and they are shared by both naive and statistically sophisticated subjects. The implications of the psychology of judgment to the analysis of rational behaviour are explored.
Pole-Seeking Brownian Motion and Bird Navigation
Authors: Kendall David G.
Published: December 2018

This paper is not concerned with identifying the mechanism of bird navigation, but rather with assessing the degree of success possible using very bad navigational aids of whatever sort. Flight-paths are simulated using virtually no hypotheses, and all numerical parameters are determined ornithologically. The results are compared with the performance of actual birds, and also with a brownian model which is analytically convenient. It turns out that some formulae for the brownian theory can be used to graduate the results obtained from actual or simulated observations, and this is its sole ornithological relevance. The data imply a surprising numerical coincidence which allows of a Darwinian explanation.

It seems likely that under suitable asymptotic hypotheses, the ornithological model will converge weakly towards the brownian one, but that question is deliberately not investigated here. Indeed it is ornithologically of little interest because (i) we are certainly not at the limit and (ii) weak convergence theorems do not normally have anything useful to say about the rate of convergence. This suggests that there may be a general need for a shift of emphasis in our thinking, away from weak convergence theorems towards weak convergence approximations (validated by simulation).
A Generalized Dirichlet Distribution in Bayesian Life Testing
Authors: Lochner Robert H.
Published: December 2018

Let F(x) be the c.d.f. for some population of life times. For real numbers t  1<t  2<…0 and F(tk)<l, let p  1 = F(t  l) and pi = F(ti) – F(ti  –1) for i = 2, 3,…, k. A generalized Dirichlet prior distribution for p = (pl …,pk) having density function f(p)∝Πi=1kpiαi−1(1−p1−…−pi)γi is considered. An intuitive justification for using this density is given and its properties are studied. Comparisons are made between this distribution and the Dirichlet distribution. The posterior distribution of F(ti) is also considered.
Model Building and the Analysis of Spatial Pattern in Human Geography
Authors: Cliff A. D. , Ord J. K.
Published: December 2018

1. Introduction and Summary

It is the purpose of this paper to determine how far various statistical models and methods of statistical inference have enabled the aims of geographical research to be met in the problem areas to which they have been applied. In so doing, we hope we can indicate to the statistician questions of geographical interest which cannot readily be answered by existing statistical methods; and to the geographer, some of the insights into geographical processes which may be gained from a statistical and model building approach. We would stress that we have not tried to be all inclusive in our coverage. Instead, we have tried to select some topics which best seem to convey the flavour of the kinds of things human geographers have been doing, and which will, at the same time, be of interest to statisticians on either theoretical or empirical grounds. In addition, our own interests mean that we have concentrated upon examples in human (economic and urban), rather than physical, geography, although similar approaches have been used there to a lesser degree. More general reviews are provided by Gould (1969), Berry (1971) and Wilson (1972).
Statistics of Directional Data
Authors: Mardia K. V.
Published: December 2018

Directional data analysis is emerging as an important area of statistics. Within the past two decades, various new techniques have appeared, mostly to meet the needs of scientific workers dealing with directional data. The paper first introduces the two basic models for the multi-dimensional case known as the von Mises–Fisher distribution and the Bingham distribution. Their sampling distribution theory depends heavily on the isotropic case and some developments are discussed. An optimum property of an important test for the von Mises–Fisher case is established. A non-parametric test is proposed for the hypothesis of independence for observations on a torus. In addition to some numerical examples on the preceding topics, five case studies are given which illuminate the power of this new methodology. The case studies are concerned with cancer research, origins of comets, arrival times of patients, navigational problems and biological rhythms. Some unsolved problems are also indicated.
Matrix Weighted Averages and Posterior Bounds
Authors: Chamberlain Gary , Leamer Edward E.
Published: December 2018

Bounds for matrix weighted averages of pairs of vectors are presented. The weight matrices are constrained to certain classes suggested by the Bayesian analysis of the linear regression model and the multivariate normal model. The bounds identify the region within which the posterior location vector must lie if the prior comes from a certain class of priors.
Measures and Models for Angular Correlation and Angular–Linear Correlation
Authors: Johnson Richard A. , Wehrly Thomas
Published: December 2018

Population models for dependence between two angular measurements and for dependence between an angular and a linear observation are proposed. The method of canonical correlations first leads to new population and sample measures of dependence in this latter situation. An example relating wind direction to the level of a pollutant is given. Next, applied to pairs of angular measurements, the method yields previously proposed sample measures in some special cases and a new sample measure in general.
Block Designs for Fractional Factorial Experiments with Replication
Authors: Cotter Sarah C.
Published: December 2018

This paper describes how incomplete block designs may be used to provide blocking plans for fractions of symmetrical factorial experiments. These designs introduce some replication, but provide a wide range of block sizes, from two units per block upwards. The blocking does not alter the aliasing of the design, and all estimable factorial effects are partially confounded, their estimates being mutually orthogonal. The analysis of these designs is described.
Super-Slowly Varying Functions in Extreme Value Theory
Authors: Anderson C. W.
Published: December 2018

Large-deviation limit theorems for maxima of random samples in the domain of attraction of the extreme value distribution Φα are studied, motivated by examination of an approximation used in the statistical analysis of large observations. The large-deviation property is shown to be equivalent to a strengthened form of slow variation condition, called here super-slow variation, on the tail of the distribution of a single observation. Various conditions for super-slow variation are discussed and Karamata's representation for slowly varying functions is extended to a wide class of super-slowly varying functions.
Conditional Independence in Statistical Theory
Authors: Dawid A. P.
Published: December 2018

Some simple heuristic properties of conditional independence are shown to form a conceptual framework for much of the theory of statistical inference. This framework is illustrated by an examination of the rôle of conditional independence in several diverse areas of the field of statistics. Topics covered include sufficiency and ancillarity, parameter identification, causal inference, prediction sufficiency, data selection mechanisms, invariant statistical models and a subjectivist approach to model-building.
The Use of Chi-Squared Statistics for Categorical Data Problems
Authors: Fienberg Stephen E.
Published: December 2018

The use of χ2 statistics for categorical data problems was initiated by Karl Pearson, but it took several years before the asymptotic distribution of these statistics was well understood. The general structure of asymptotic results for χ2 statistics is reviewed and the applicability of the general structure to a variety of problems of practical interest is discussed. These problems include the use of χ2 statistics in small-sample situations and in large sparse tables, in cluster sampling, and in cases where they do not have asymptotic χ2 distributions.
A Note on Dissociatedness and Order Statistics
Authors: McGinley W. G.
Published: December 2018

Using the notion of dissociatedness (McGinley and Sibson, 1974) we prove the almost sure convergence of some convex combinations of order statistics and a special case of our result is shown to relate to the convergence of Gini's coefficient of mean difference.
Bandit Processes and Dynamic Allocation Indices
Authors: Gittins J. C.
Published: December 2018

The paper aims to give a unified account of the central concepts in recent work on bandit processes and dynamic allocation indices; to show how these reduce some previously intractable problems to the problem of calculating such indices; and to describe how these calculations may be carried out. Applications to stochastic scheduling, sequential clinical trials and a class of search problems are discussed.
The Multiparameter Steady Model
Authors: Smith J. Q.
Published: December 2018

This paper generalizes and extends the idea of a steady evolution of a time series to multiparameter processes with any observational distribution. Particular examples of such processes are given. These include the multivariate steady process, univariate normal processes where the observational variance is unknown and series of multinomial observations with a steady model on the cell probabilities.
Equivalent Forms of the Multivariate Portmanteau Statistic
Authors: Hosking J. R. M.
Published: December 2018

The Box-Pierce portmanteau test for univariate autoregressive moving-average (arma) time-series models has been generalized to multivariate time-series models by several authors, and several different definitions of the test statistic have resulted. The purpose of this note is to point out that all these different forms of the portmanteau statistic are in fact identical.
On a System of Two-Dimensional Recurrence Equations
Authors: Besag Julian
Published: December 2018

The solution of a system of two-dimensional recurrence equations is discussed, particularly in the context of the auto-correlation structure of a stationary first-order auto-normal scheme on the doubly-infinite rectangular lattice. A simple approximation in the region of primary interest is described.
Comments and Queries a Note on Some Results of Tamhane
Authors: Lam K. , Mehra K. L.
Published: December 2018

The relative efficiency of the matched sample design in selecting the better of the two Bernoulli treatments is considered. Exact conditions for the matched sample design to have efficiency equal to a half and one are given respectively.
On Linear Intensity Models for Mixed Doubly Stochastic Poisson and Self-Exciting Point Processes
Authors: Ogata Yosihiko , Akaike Hirotugu
Published: December 2018

A flexible family of parametric models for intensity processes is introduced to represent a causal relationship between a point process and another stochastic process. Algorithms for the maximum likelihood computation and the procedure of model selection are discussed.
Approximate Posterior Distributions for Incomplete Data Problems
Authors: Laird Nan M. , Louis Thomas A.
Published: December 2018

We consider the problem of developing a simple approximation to a posterior distribution arising from incomplete data sampling. We compare approximations based on the normal distribution and upon conjugate distributions, theoretically where feasible and in some numerical examples. Two general conclusions can be made on the basis of our numerical work. First, when there is missing data, approximations which fail to take loss of information into account give overly concentrated posterior distributions. Second, the Gaussian approximation matching mode and observed Fisher information is quite good with large sample sizes and true posteriors which are not highly skewed. Further work delimiting these conditions more precisely will be useful. Finally, the use of a conjugate posterior which matches both mode and information performs well in all of our examples, and is decidedly superior to the Gaussian approximation with small samples.
Definitions and Properties for M-Concurrence Designs
Authors: Jarrett R. G.
Published: December 2018

Incomplete block designs can be regarded as being in some sense approximately partially balanced. In this paper, we show that parameters pjki corresponding to those used for partially balanced designs can be defined for certain designs with m distinct concurrences. These parameters have properties analogous to those for m associate-class partially balanced incomplete block (PBIB(m)) designs and have proved useful in searching for optimal designs. This leads to a new upper bound for the efficiency factor which, for given pjki values, is attained if a PBIB(2) design exists. Some results on complement and dual designs are also given.
Statistical Uses of the Spatial Median
Authors: Brown B. M.
Published: December 2018

The spatial median, called the mediancentre in an earlier paper by J. C. Gower, is defined as the bivariate location measure to minimize the sum of absolute distances to observations. its asymptotic efficiency relative to the sample mean vector with normal data is shown to exceed the usual univariate 2/π = 0.637. its estimating equations have an angular aspect and are used to develop “angle tests”, which are analogues of sign tests in one dimension.
On the Convergence of the EM Algorithm
Authors: Boyles Russell A.
Published: December 2018

An example is given showing that a sequence generated by a GEM algorthm need not converge under the conditions stated in Dempster et al., (1977). Two general convergence results are presented which suggest that in practice a GEM sequence will converge to a compact connected set of local maxima of the likelihood function; this limit set may or may not consist of a single point.
Nearest Neighbour (Nn) Analysis of Field Experiments
Authors: Wilkinson G. N. , Eckert S. R. , Hancock T. W. , Mayo O.
Published: December 2018

The paper is in two parts. Part I presents results of a Monte Carlo randomization study of Papadakis's covariance method of NN analysis which show that (i) a non-iterated Papadakis analysis tends to be conservatively biassed; (ii) iteration of the analysis as suggested by Bartlett (1978) leads to substantial positive bias in the treatment F ratio; (iii) the method is very inefficient when there are substantial trend effects in the data. A theoretical explanation of these results is given.

Part II describes a new method of NN analysis discovered by the first author and developed in collaboration with the co-authors. The method is essentially a “moving-block” analogue of classical forms of analysis for “fixed” blocks (or rows, columns). It avoids the defects of Papadakis's method and leads to approximately unbiassed analyses. It is nearly always and often substantially more efficient on average than classical analyses of complete or incomplete block experiments, and also more efficient than standard analyses of Latin or lattice square designs if there are appreciable row x column interactions in the data. New criteria of design for NN balance are described. Validity of the new method under randomization is demonstrated empirically with Monte Carlo studies.
Semi-Latin Squares and Related Designs
Authors: Preece D. A. , Freeman G. H.
Published: December 2018

A semi-Latin square for t = kn symbols is a rectangular arrangement with n rows and t columns, these latter being grouped into sets each containing k consecutive columns; each symbol occurs exactly once in each row and exactly once in each set of columns (Yates, 1935). Previous authors seem not to have considered the combinatorial possibilities of semi-Latin squares. For each pair of values k and t, the semi-Latin squares can be classified into “species” or “main classes”, akin to the species of Latin squares. The classification considered in this note disregards the ordering of the k symbols to be found where any row intersects any set of columns; for k = 2 and t = 4, 6 and 8, the numbers of species are 1, 2 and 10 respectively. The relationship between semi-Latin squares, Trojan squares and certain partially balanced incomplete block designs with two associate classes is discussed. The relevant semi-regular designs of Clatworthy (1973) are examined to see which can be obtained from semi-Latin (Trojan) squares, and some enumerations of squares obtainable from the semi-regular designs are reported.
Approximating Priors by Mixtures of Natural Conjugate Priors
Authors: Dalal S. R. , Hall W. J.
Published: December 2018

The purpose of this paper is twofold: First, to show that a natural enlargement of the class of natural conjugate priors–namely, mixtures of natural conjugate priors–also leads to mathematically tractable solutions. Secondly, to show that this enlargement is “adequate” in that any prior may be arbitrarily closely approximated by a suitable member of this class.

Specifically, a general method for approximating an arbitrary prior density is first given with approximation (convergence) defined pointwise or in the L  1 or total variation senses. Corresponding posterior densities are shown to provide approximations in corresponding senses, whether or not the approximating priors are those given herein. Then formulas are given for posterior densities when the prior is a mixture of natural conjugate priors, and it is shown that an arbitrary prior density and its posterior density may be approximated by such a mixture and its posterior. In these senses, Bayesians may find it satisfactory to confine attention to mixtures of natural conjugate priors. The results are illustrated by several exponential family examples.
Regression, Prediction and Shrinkage
Authors: Copas J. B.
Published: December 2018

The fit of a regression predictor to new data is nearly always worse than its fit to the original data. Anticipating this shrinkage leads to Stein-type predictors which, under certain assumptions, give a uniformly lower prediction mean squared error than least squares. Shrinkage can be particularly marked when stepwise fitting is used: the shrinkage is then closer to that expected of the full regression rather than of the subset regression actually fitted. Preshrunk predictors for selected subsets are proposed and tested on a number of practical examples. Both multiple and binary (logistic) regression models are considered.
Note on Large Sample Properties of Balanced Samples
Authors: Pfeffermann D.
Published: December 2018

Assuming the general linear “super population” model, the optimal predictor of the finite population mean is shown to converge in probability to the expansion estimator, for general structures of the variance—covariance matrix, if the sample is balanced. The implications of this result are demonstrated through various simple examples.
A Goodness-Of-Fit Test for the von Mises–Fisher Distribution
Authors: Mardia K. V. , Holmes D. , Kent J. T.
Published: December 2018

We derive a goodness-of-fit test for the von Mises-Fisher distribution which is an extension of a goodness-of-fit test for the von Mises distribution given by Cox (1975). A robust version of the test is also given. Computational aspects of the test statistic are discussed and a numerical example is given.
Canonical Parameterizations and Zero Parameter-Effects Curvature
Authors: Kass Robert E.
Published: December 2018

An equation derived by Hougaard (1982) characterizes certain canonical parameterizations of curved exponential families. The geometrical interpretation of Hougaard's equation is presented and, with this background, the relationship of work on Gaussian non-linear regression, by Bates and Watts (1980) and others, to that on general asymptotic inference, by Efron (1975) and Amari (1982a, b), is discussed. Hougaard's δ-parameterization is that in which there is zero “parameter-effects curvature” with respect to Amari's (1982b) α-connection geometry, when α= 1–2δ. For the generalized linear models of Nelder and Wedderburn (1972) there is a separate “parameter-effects” array for each α, and each is a convex combination of the exponential and mixture “parameter-effects” arrays. On the other hand, in non-linear regression, for any family of symmetric errors there is only one array and it is a constant multiple of that for the Gaussian family.
Upper Bounds for Factorial Efficiency Factors
Authors: Lewis S. M. , Dean A. M.
Published: December 2018

The use of the upper bound for the average efficiency factor of a 2-factor interaction derived by Dean and Lewis (1983) is extended to the case where neither main effect achieves full efficiency. It is shown how the resulting boundary surface can be used to assess a design and hence to aid the construction of efficient designs. The bound is also generalised to higher order interactions.
Schläflian Rotatability
Authors: Draper Norman R.
Published: December 2018

Suppose that a polynomial of order d is to be fitted to data from a designed experiment and that V(·) is any selected variance function. For example, if ŷ(x) is the fitted value at a location x = (x  1, x  2, …, xk)′ in the k-dimensional predictor space, V{ŷ(x)}, which we shall also write V(ŷ), denotes the usual variance function. Design rotatability of order d is the property that V{ŷ(x)} has spherical contours in the x space. Here, a more general definition, Schläflian rotatability, is made with respect to any V(·), by requiring that the contours of F(·) are spherical in the space of the model predictors, specified in the usual Schläflian form. It is shown that Schläflian rotatability for V(ŷ) implies ordinary rotatability, and vice versa. The selection of a particular Schläflian variance profile for V(·) for a second order rotatable design implies a choice of the number of center points and the consequences of this are explored.
Optimal Sequential Estimation for Ergodic Birth-Death Processes
Authors: Manjunath S. M.
Published: December 2018

For ergodic birth-death processes a lower bound for the variance of an unbiased estimator of a function of birth and death parameters is obtained under an arbitrary stopping rule. All efficiently estimable functions and closed efficient sampling schemes are characterized. The processes treated here include random walk with reflecting barriers, immigration-death process, M/M/1 and M/M/S queues with limited or unlimited waiting space.
Some Aspects of the Spline Smoothing Approach to Non-Parametric Regression Curve Fitting
Authors: Silverman B. W.
Published: December 2018

Non-parametric regression using cubic splines is an attractive, flexible and widely-applicable approach to curve estimation. Although the basic idea was formulated many years ago, the method is not as widely known or adopted as perhaps it should be. The topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology. Particular subjects covered include the basic principles of the method; the relation with moving average and other smoothing methods; the automatic choice of the amount of smoothing; and the use of residuals for diagnostic checking and model adaptation. The question of providing inference regions for curves – and for relevant properties of curves – is approached via a finite-dimensional Bayesian formulation.
Gaussian Estimation for Correlated Binomial Data
Authors: Crowder Martin
Published: December 2018

The mean/variance structure of correlated binomial data is discussed, and Whittle's (1961) Gaussian estimation is suggested as a useful general method in this context. Comparisons are made with other moment methods in current use.
Modelling Data Containing Exact Zeroes Using Zero Degrees of Freedom
Authors: Siegel Andrew F.
Published: December 2018

The noncentral chi-squared distribution with zero degrees of freedom can be used to model continuous data of a variety of shapes that also contain exact zero values. Existence and uniqueness of the maximum likelihood estimates for the scaled non-central chi-squared distribution with zero degrees of freedom are established, computational methods are considered, and an illustrative example is given.