bluuebunny/crossref_metadata_2025 huggingface dataset:
Power of Tukey's Test for Non-Additivity
Authors: Ghosh M. N. , Sharma Divakar
Published: December 2018

The problem of testing for non-additivity in a two-way classification has been considered in this paper where the mean μii can be expressed as μ+αi+βi where αi and βi are the effects of the two ways of classification (i = 1, …,p;j = 1, …,q). Tukey (1949) has suggested a test for non-additivity. The power function of this test has been found in this paper for alternatives of the form μii = μ+αi+βi+cαi βi and numerical calculations made for the case of p = q = 6. For the sake of comparison the special case of α2i–1 = α2i, β2j–1 = β2j has also been considered, when an F-test is available with d.f. (4, 27). It is seen that the power of Tukey's statistic is slightly less than that of the F-test for smaller values of C  2 σ2 and ∑αi2σ2=∑βj2σ2, but for a wide range of values of these parameters, the performance of Tukey's statistic is better.
Higher Moments of a Maximum-Likelihood Estimate
Authors: Shenton L. R. , Bowman K.
Published: December 2018

For a distribution depending on a single parameter the first four sampling moments of the maximum-likelihood estimate to orders N−2, N−3, N−3 and N−4 respectively are given. Expressions for the measures of skewness γ1 and γ2 are also given. Several illustrative examples are included as a check on the heavy algebra. The paper extends earlier work by Haldane and Smith.
Higher-Order Properties of a Stationary Point Process
Authors: McFadden J. A. , Weissblum Walter
Published: December 2018

This paper is concerned with joint distributions, joint moments, and joint renewal functions in a stationary point process. The random variables considered are (a) the numbers of events in adjacent intervals of given lengths, located either arbitrarily in time or by an arbitrary event, and (b) the lengths of intervals measured to the various events, beginning either at an arbitrary point or at an arbitrary event. The different bivariate probability functions are related to each other by means of transforms. Moments are considered. In particular, the marginal moments of the number of events, counted by either method, are related to transforms of the multivariate renewal functions. Each product moment of the interval lengths is expressed as a multiple integral of the joint probability of certain numbers of events. A new theorem is proved, more general than in a previous paper, concerning the superposition of two independent stationary point processes. This theorem gives conditions under which the successive intervals between events can be independent in each of the constituent processes and also in the merged process.
An Approximate Solution of the Integral Equation of Renewal Theory
Authors: Bartholomew D. J.
Published: December 2018

An approximation is given to the solution of the integral equation for the renewal density. The approximation is of particular value when the renewal distribution has a high degree of positive skewness. For an important class of such distributions it also provides an upper bound to the renewal density. Some numerical comparisons are given for renewal processes in discrete time. Other methods of obtaining an approximate solution to the equation are briefly reviewed and compared with the new method.
A Test for Variance Heterogeneity in the Residuals of a Gaussian Moving Average
Authors: Herbst Laurence N.
Published: December 2018

A large sample test for variance heterogeneity in the residuals of a Gaussian moving average is presented, based upon periodogram-ratios of the data. For long series, it is shown that neither the null-hypothesis nor alternative-hypothesis behaviour of the test statistic is dependent upon the coefficients of the moving average (and these need not be estimated in advance). The alternative-hypothesis behaviour of the test statistic is discussed in terms of the coefficients in a Fourier expansion of the non-constant variance function.
On the Asymptotic Behaviour of Queues
Authors: Keilson J.
Published: December 2018

The homogeneous process N(t) on the lattice of integers in continuous time is discussed. When the increment probability generating function ϵ(u) is analytic at u = 1, the transition functions γn(t) = p{N(t) = n |  N(0) = 0} are Bessel-like functions with simple asymptotic properties. The asymptotic behaviour of a variety of M/G/1, G/M/1 and related queueing systems may be deduced from these properties. An appendix is devoted to a review of saddle point methods and central limit behaviour for processes in continuous time.
Further Models for Estimating Correlation in Discrete Data
Authors: Tallis G. M.
Published: December 2018

This note considers the distribution of the sum of n identically distributed multinomial variates Xi, the correlation coefficient for Xi and Xj being ρ for all i, j, i ≠ j. The initial model, with n and p' = (p1, p2,…,pk) fixed, is generalized by allowing first n, then p and finally both n and p to be random variables.
Relationships between Bayesian and Confidence Limits for Predictions
Authors: Thatcher A. R.
Published: December 2018

Given the number of successes in a random sample, prediction limits can be determined for the number which will be observed in a second sample, in a way which does not depend on any assumption or inference about the unknown proportion in the population. Such “confidence limits” for the prediction are found to correspond to Bayesian solutions based on two particular prior distributions, and are related to Laplace's rule of succession. The results suggest a possible type of “prediction strategy”.
Note on Interval Estimation in Non-Linear Regression When Responses are Correlated
Authors: Halperin Max
Published: December 2018

Recent papers by Williams and Halperin discuss procedures for obtaining confidence regions† for parameters in non-linear regression. These results assume, among other things, independence and homoscedasticity of variance of the dependent variables. This note points out that by using some results of Rao it is possible to drop these latter assumptions.
A Proposed Two-Sample Rank Test: The Psi Test and its Properties
Authors: Gibbons Jean D.
Published: December 2018

Since the power of the Psi test proposed by Gibbons (1964) compares favourably with that of other one-sided, two-sample rank tests on the equality of two distribution functions, its properties merit further investigation. This paper is devoted to such a study. A table of exact critical values for the test statistic is given for selected small sample sizes, and compared with critical values for the approximate distribution which is a function of Student's t. The agreement is good even for small samples. Using the results of Chernoff and Savage (1958), the asymptotic distribution is found to be normal in both the null and alternative cases. The null mean and variance are 0 and rπ  2l(3mN) respectively. Expressions are given for the alternative moments. The relation of the Psi test to other test statistics, specifically Terry's c  1 test, the Wilcoxon or Mann-Whitney test, and Savage's DN statistic, is investigated in terms of correlation coefficients in the null case. The test is most similar to Terry's c  1 statistic.
Some Statistical Problems Connected with Crystal Lattices
Authors: Domb C.
Published: December 2018

Three statistical problems are discussed which are of importance in physics. The first arises from the statistical mechanics of interacting systems on crystal lattices, and the statistical problem concerns the distribution of nearest neighbour contacts when two different species randomly occupy the sites of a crystal lattice. The relation between the partition function in statistical mechanics and the characteristic function in statistics is investigated. The second problem concerns the mean cluster size in such a random mixture, and a feature of particular interest is the critical concentration at which there is a non-zero probability of clusters stretching right across the crystal. The third problem deals with the properties of self-avoiding walks on lattices, such as their total number, mean square length, and resultant space distribution.
Confidence-Region Tests
Authors: Aitchison J.
Published: December 2018

To test a hypothesis that a parameter has value θ* we usually ask whether an observation falls in a critical region of the outcome space. It is well known that, by a suitable choice of confidence region, an equivalent test is to ask whether θ* lies outside the confidence region. In this paper the concept of such a confidence-region test is extended in directions which are relevant to the study of data-snooping and multiple-hypothesis testing. These problems are investigated for linear models and, in the asymptotic case, for more general parametric models.
On Confidence Points and Bayesian Probability Points in the Case of Several Parameters
Authors: Peers H. W.
Published: December 2018

The problem of formal equivalence between confidence points and Bayesian probability points in situations where there are several unknown parameters is discussed, and the relevant asymptotic formulae are derived to O(n  –½).
Polynomial Projecting Properties of Multi-Term Predictors/Controllers in Non-Stationary Time Series
Authors: Ray W. D. , Wyld C.
Published: December 2018

Box and Jenkins (1962, 1963) introduce a practical predictor for handling non-stationary types of time series and they show it to be optimal for a particular class of stochastic processes. As a consequence the predictor can be shown to follow stochastic as well as deterministic polynomial trends in the series. The present paper sets out to evaluate the bias term in the expression for the error when applying this predictor, and shows how the bias can be reduced sometimes to zero for particular values of the parameters. This determination of bias is done in the context of deterministic polynomial trends but it can be shown to apply to the stochastic case also.
The Statistical Fourier Analysis of Variances
Authors: Herbst L. J.
Published: December 2018

Let Xt (t = 1, 2, …, N) denote a series of independent normal random variables, with zero means and non-zero finite variances ft (t = 1, 2, …, N). Then ft and ft−1 admit finite Fourier expansions. The present paper is a study of the statistics N−1∑t=1NXt2cos (2πts/N) and N−1∑t=1NXt2sin (2πts/N) as estimators of the sth Fourier coefficients of ft, and of the distributional properties of suitable ratios of these statistics as (sufficient) test statistics for non-constancy of ft, directed at alternatives where the Fourier expansion of ft−1 has a simple form.
A Two-Dimensional Poisson Growth Process
Authors: Morgan R. W. , Welsh D. J. A.
Published: December 2018

A study is made of the rate at which infection travels through the points of a lattice when the mechanism of a spread is a Poisson process. The theory developed is compared with the results obtained by a Monte Carlo experiment.
Recursive Relations for Predictors of Non-Stationary Processes
Authors: Whittle P.
Published: December 2018

It is shown that if a process {xt} is generated recursively by a relation of type ∑j=0pajtxt−j=∑k=0qbktϵt−k+λt, where ajt, bkt and λt, are time-dependent sequences and {ϵt} is a process of unautocorrelated variables, then in general the linear least-square predictor of xt+m based upon {xs; s≤t}, denoted vt, obeys a recursion of the type ∑k=0qβktξt−k=∑j=0τcjtxt−j+vt, where r = max(p – 1, q – m). A method is given for obtaining the coefficients βkt, cjt and vt, and this solution is made explicit for various special cases.
Bayesian Estimation of Parameters of a Multivariate Normal Distribution
Authors: Evans I. G.
Published: December 2018

A significant omission from the work of Raiffa and Schlaifer (1961) relating to a multivariate normal distribution is consideration of the case when the dispersion matrix is completely unknown. This paper partially fills the gap by applying the methods of Raiffa and Schlaifer to the particular problems of estimating the mean μ, the dispersion matrix ∑, and the generalized variance | ∑ |, when either or both of μ and ∑ are unknown.
The Basic Birth-Death Model for Microbial Infections
Authors: Williams T.
Published: December 2018

On the assumption that parasite populations in vivo are governed by a linear, homogeneous birth-death process with undistributed parameters and that they generate only marginally faster than the host can destroy them, the distribution of their number at any time is obtained in terms of a Bessel function of imaginary argument. From this is derived the distribution of incubation period, defined as the first-passage time to a high response threshold. Various properties of the model are noted and compared with known characteristics of the infectious process.
Quasi-Stationary Distributions and Time-Reversion in Genetics
Authors: Seneta E.
Published: December 2018

Finite absorbing Markov chains appear frequently as models for genetic processes. Although in such chains absorption in a finite time is certain, this time may be long, and it is then of interest to consider quasi-stationary distributions as describing the process in such situations; this is also true of certain infinite chains. Two such distributions are discussed, and derived for three particular genetic models as diffusion approximations. The distribution regarded as more satisfactory has interesting properties as regards “time-reversion” of the genetic process. A general discussion of this concept leads to the quasi-stationary generalization of Wright's formula for stationary distributions.
An Approach to the Study of Markov Processes
Authors: Kingman J. F. C.
Published: December 2018

Whenever random processes are studied or applied, Markov processes almost inevitably appear. This paper looks at such processes from the point of view of an observer who takes note of the process only when it is in one particular state. This apparently perverse standpoint has yielded a powerful tool for the study of Markov processes on a discrete state space.
The Reliability of Multiplex Systems with Repair
Authors: Downton F.
Published: December 2018

This paper derives expressions for the life-time distributions of systems consisting of a number of components in parallel, where repairs to these components are possible. These distributions reduce to those of first-passage times in a birth-death process or in a semi-Markov (or Markov renewal) process, and have application outside reliability theory. In the reliability context, the effects of altering the number of components and of increasing the repair facilities are briefly compared, and a comparison of different repair distributions is made.
Some Aspects of the Estimation of Quantiles
Authors: Loynes R. M.
Published: December 2018

The problem of estimating quantiles in the context where restriction to particular parametric families is inappropriate is considered, for simple random sampling and stratified sampling. For the former type of sampling the distribution-free procedures are obtained and the admissible estimators identified, and certain other questions are dealt with. In the stratified sampling situation similar problems are considered, less successfully.
Balanced Factorial Designs
Authors: Kshirsagar A. M.
Published: December 2018

Kurkjian and Zelen (1963) have proved that factorial designs possessing a particular structure for their C matrix (they call this property A) are factorially balanced. This paper proves the converse of this result, viz. factorially balanced designs are (a) EGD/(2n - 1)-PBIB designs as defined by Hinkelmann (1964), and (b) possess property A. The latent roots and vectors of the C matrix of such designs are obtained in a much simpler form than that given by Hinkelmann.
Optimization of a Hot Rolling Mill
Authors: Dowson D. C.
Published: December 2018

This paper is concerned with constructing rules for determining the times at which slabs have to be fed into successive stages of a hot rolling mill so as to optimize the working of the mill. The errors entering the system are assumed to form stationary stochastic sequences and the method adopted for determining the appropriate feedback is that of linear least squares. The analysis is carried through and results tabulated for the case of uncorrelated errors and also for one, fairly general, case of correlated errors.
Optimum Utilization of Auxiliary Information: πPs Sampling of Two Units from a Stratum
Authors: Hanurav T. V.
Published: December 2018

For the estimation of the population total of a character y, defined over a finite population of known size, “optimum” sampling strategies for the case when auxiliary information X of a type commonly met with in practice are recalled. A method of sampling two units from a stratum such that the probability of inclusion of one unit is exactly proportional to the value of X on the unit is given. This method incidentally furnishes not only an optimum estimator of the population total but also a stable non-negative unbiased estimator of the variance of the main estimator. The method is compared with two other methods in varying probability sampling. Illustrative examples are given.
Considering Statistical and Time Averages in a Regulation Problem
Authors: Kounias Eustratios
Published: December 2018

In this paper statistical and time averages are compared in a certain regulation problem, and under conditions of stationarity it is proved that in practice the difference between them is negligible. The predictors–controllers are taken as linear functions of the known data. Numerical results are presented supporting the theory.
Four-Letter Words: The Distribution of Pattern Frequencies in Ring Permutations
Authors: Barton D. E. , David F. N.
Published: December 2018

An alphabet of c letters is considered. There are, then, ck possible ordered sequences of k of these. Such k-sequences are grouped into λ mutually exclusive and exhaustive sets; members of each set (called k-patterns) are counted the same. A general expression for the joint generating function of the frequencies of occurrence of the different k-patterns is obtained when r letters, with fixed numbers of each sort, are randomly arranged in a ring. In particular, k-sequences which comprise the same set of letters, in whatever order, are called k-words. Explicit expressions are obtained for the cases k = 3, 4, c = 2 both from the generating function and by direct combinatorial arguments. The asymptotic (multinormal) limiting distribution is obtained for the general case and more explicitly for k-letter words.
The Mixed Binomial Distribution and the Posterior Distribution of p for a Continuous Prior Distribution
Authors: Hald A.
Published: December 2018

Asymptotic expansions are derived for the mixed binomial distribution and the posterior distribution of p for a continuous prior distribution. Expansions are also found for the first four moments of the posterior distribution. Applications to estimation and sampling inspection are discussed.
The Choice of the Degree of a Polynomial Model
Authors: Hager H. , Antle C.
Published: December 2018

This paper presents an empirical evaluation of several methods of estimating the degree of a polynomial model.
The Ergodic Theory of Subadditive Stochastic Processes
Authors: Kingman J. F. C.
Published: December 2018

An ergodic theory is developed for the subadditive processes introduced by Hammersley and Welsh (1965) in their study of percolation theory. This is a complete generalization of the classical law of large numbers for stationary sequences.
A Remark on Solving Equations in Sums of Powers
Authors: Searle S. R.
Published: December 2018

Equations in sums of powers that can arise in experimental design situations are readily solved by using a determinantal form of Newton's formulae for the symmetric sums of products of zeros of a polynomial.
The Choice of Variables in Multiple Regression
Authors: Lindley D. V.
Published: December 2018

This paper is concerned with the analysis of data from a multiple regression of a single variable, y, on a set of independent variables, x  1,x  2,...,xr. It is argued that the form of the analysis should depend on the use that is to be made of the regression, and that therefore an approach employing ideas from decision theory may be worth while. Two situations are analysed in this way: in the first it is desired to predict a future value of y; in the second we wish to control y at a preassigned value. The two analyses are found to be different: in particular, the standard errors of the regression coefficients are found to be irrelevant in the prediction problem, but not in the control problem. In the former it is shown that, under rather special assumptions on the multiple regression experiment, the analysis is similar to that recommended by other writers. If the costs of control do not depend on the values at which the control takes place, a similar analysis holds for the second problem. The approach throughout is Bayesian: there is no discussion of this point, I merely ask the non-Bayesian reader to examine the results and consider whether they provide sensible and practical answers.
Small Sample Power of a Non-Parametric Test for the Bivariate Two-Sample Location Problem in the Normal Case
Authors: Mardia K. V.
Published: December 2018

The exact power of a test U (Mardia, 1967) is obtained for normal location alternatives, but even for small samples the expression cannot be simplified. The empirical power function of the U-test is obtained and compared with the power of Hotelling's T2 by extending some efficiency measures and developing a new non-central T2 routine. It is found that the efficiency of the U-test relative to Hotelling's T2 test decreases with N and lies between 79 and 93 per cent.
Multiparameter Bayesian Indifference Procedures
Authors: Novick Melvin R.
Published: December 2018

The method of minimum necessary samples formulated by Novick and Hall from a suggestion by Lindley is refined and extended to cover multiparameter models. A number of such models are studied and Bayesian indifference specifications are obtained for each within extended natural conjugate families. In all cases the one-parameter conditional specifications derived from the multiparameter specifications are consistent with specifications derivable from corresponding one-parameter models.

Models studied include the multinomial models, one- and two-parameter uniform models, an exponential model with a range parameter, and the two-parameter normal model. For these models the specifications seem entirely straightforward.

Indifference specifications are also obtained for the simple regression model and the bivariate normal model. It is noted that these specifications do not agree but the relationship between these two specifications is described and in part justified. It is pointed out that these differences could, in fact, mirror important differences in the structural assumptions of these models. A random effects model of value in mental test theory is studied briefly. This model provides a framework for a discussion of the multiple comparisons problem. Finally, some summary comments on the methods of this paper are offered.
Some Non-Orthogonal Fractions of 2N Designs
Authors: John Peter W. M.
Published: December 2018

This paper is concerned with the class of fractions obtained when a 2n-k fraction is omitted from a complete 2n factorial. The three-quarter replicate is a special case. It is shown that the maximum-likelihood estimates of effects are the averages of estimates from over-lapping half replicates. Examples are given, including a seven-eighth fraction of a 25 design of resolution VII, a design of resolution VI for eight factors in 112 points and a design of resolution V for eleven factors in 96 points.
A Sequential Procedure for Testing a Null Hypothesis Against a Two-Sided Alternative Hypothesis
Authors: Billard L. , Vagholkar M. K.
Published: December 2018

A new sequential test procedure is introduced and compared with an existing procedure for testing a null hypothesis against a two-sided alternative hypothesis.
A Note on the Posterior Mean of a Population Mean
Authors: Ericson W. A.
Published: December 2018

It is a well-known result, see for example Lindley (1965) and Raiffa and Schlaifer (1961), that if x̄ is the mean of a sample of independent observations distributed N(μ, σ2) where σ2 is known, and if μ has been assigned a normal prior distribution, N(m, v), then the posterior expectation of μ, given the sufficient statistic x̄, has the form {x̄(n/σ2) + m/v}/{(n/σ2)+1/v}, that is, has the intuitively appealing form of a weighted average of the prior mean and sample mean with weights inversely proportional to the prior variance and the conditional sampling variance of X̄ respectively. In this note we present an extremely simple generalization of this result which seems not to have been noted explicitly before and which holds for a variety of distributions other than the normal.
Testing for Linear Contagion—Inverse Sampling
Authors: Peritz E. , Shlom E.
Published: December 2018

Let Tr (r = 0, 1, …) be independent, positive, exponentially distributed random variables with parameters λ + rΨ. It is shown that the uniformly most powerful unbiased test of the null hypothesis Ψ = 0 against alternatives Ψ > 0 rejects the former for small values of Z=(∑r=1nrTr)(∑r=0nTr)−1. The power function of the test is given and the test is shown to be consistent. Unbiasedness is shown for a wider class of alternatives. A sequential version of the test is presented. A generalization of the results for a model relevant to accident research is given in the last section of the paper. Selected values of the power function are tabulated in the Appendix.
Asymptotic Properties of Bayesian Decision Rules for Two Terminal Decisions and Multiple Sampling. I
Authors: Hald A. , Keiding N.
Published: December 2018

The model is based on a differentiable prior distribution, a linear loss function, an asymptotically normal sampling distribution and sampling costs proportional to the sample size. Asymptotic expressions are derived for sample sizes, acceptance and rejection criteria and the minimum regret by minimizing the average regret for a k-stage sampling and decision procedure. The results generalize previously known results for k = 1 and 2 and bridge the gap to sequential sampling.
Correlation between Two Vector Variables
Authors: Kshirsagar A. M.
Published: December 2018

Ruben (1966) has suggested a simple approximate normalization for the correlation coefficient in normal samples, by representing it as the ratio of a linear combination of a standard normal variable and a χ variable to an independent χ variable and then using Fisher's approximation to a χ variable. This result is extended in this paper to a matrix, which in a sense is the correlation coefficient between two vector variables x and y. The result is then used to obtain large sample null and non-null (but in the linear case) distributions of the Hotelling–Lawley criterion and the Pillai criterion in multivariate analysis. Williams (1955) and Bartlett (1951) have derived some exact tests for the goodness of fit of a single hypothetical function to bring out adequately the entire relationship between two vectors x and y, by factorizing Wilks's A suitably. These factors are known as “direction” and “collinearity” factors, as they refer to the direction and collinearity aspects of the null hypothesis. In this paper, the other two criteria viz. the Hotelling-Lawley and Pillai criteria are partitioned into direction and collinearity parts and large sample tests corresponding to them are derived for testing the goodness of fit of an assigned function.
Use of the Kolmogorov–Smirnov, Cramér–Von Mises and Related Statistics Without Extensive Tables
Authors: Stephens M. A.
Published: December 2018

This paper gives modifications of eleven statistics, usually used for goodness of fit, so as to dispense with the usual tables of percentage points. Some test situations are illustrated, and formulae given for calculating significance levels.
Sampling with Random Stratum Boundaries
Authors: Fuller Wayne A.
Published: December 2018

Sampling designs are presented which admit unbiased estimators of the variance of the mean and which possess efficiency for the mean approximately equal to 1 per stratum. The designs are developed for unequal as well as equal probability sampling. Improved estimators of the variance of the estimated mean are considered.
A Method for Discriminating between Models
Authors: Atkinson A. C.
Published: December 2018

It is desired to determine which of several alternative models adequately describe the data. The properties of a combined distribution containing the component models as special cases are investigated. Using this distribution, statistics are developed for testing for departures from one model in the direction of another and for testing the hypothesis that all models fit the data equally well. The relationship with other procedures is investigated. Examples are given of the use of the method, especially when there are two component models belonging to separate parametric families.
The Measurement of Association in a Contingency Table: Three Extensions of the Cross-Ratios and Metrics Methods
Authors: Altham Patricia M. E.
Published: December 2018

The methods of cross-ratios, equivalence classes and metrics that were introduced in a previous paper (Altham 1970) are extended to three further problems. First the interdependence of the rows, columns and layers of a three-way contingency table is discussed. There are essentially four different types of association to be considered: the complete dependence of rows, columns and layers, the dependence of rows and columns conditional on a given layer, the dependence of the two classifications “rows-and-columns” and layers, and the second-order interaction of rows, columns and layers. The relations between these four types of association are discussed in terms of the appropriate equivalence classes. In order to obtain measures of dependence of rows, columns and layers, metrics are defined on the equivalence classes. Some examples of metrics are given. The consideration of three-way tables is concluded by a discussion of the asymptotic posterior distributions of certain special metrics, which are suggested by the work of Lindley (1964), and Plackett (1962).

Cross-ratios methods are also extended to the problem of measuring association in a two-way table with some cells deleted. The paper is concluded by consideration of the problem of comparing associations in two distinct contingency tables, some of whose cells may be deleted.
Posterior Precision for Non-Normal Distributions
Authors: Finucan H. M.
Published: December 2018

The result “average posterior precision equals or exceeds prior precision plus average observation precision” is proved for any distributions of a single variable. The “average” used is in effect a harmonic mean.
The Present State of Multiple Comparison Methods
Authors: O'Neill R. , Wetherill G. B.
Published: December 2018

In this paper we discuss the problems, formulations and proposed solutions in the area of multiple comparisons. A classified bibliography is also given.
On Existence and Mixing Results for Cluster Point Processes
Authors: Westcott M.
Published: December 2018

Cluster point processes are defined and studied using the probability generating functional. Necessary and sufficient conditions for the existence of a cluster process are proved and applied to particular cases. A result on mixing in cluster processes is established.
The Efficient Use of Supplementary Information in Standard Sampling Procedures
Authors: Foreman E. K. , Brewer K. R. W.
Published: December 2018

Use is made of a well-known heteroscedastic super-population model to obtain comparisons of the efficiencies of six methods of sampling in common use. These are simple random sampling with unbiased estimators, simple random sampling with ratio estimators, and sampling with probability proportional to size with unbiased estimators; each with and without replacement. The resulting comparisons are readily interpretable, thus affording simple criteria for comparing the efficiencies of these widely used sample plans. They also provide heuristic explanations of their properties.
The Covariance Structure of the Departure Process from M/G/1 Queues with Finite Waiting Lines
Authors: King Ralph A.
Published: December 2018

This paper contains a study of the departure process from M/G/1 queues with a waiting space of size N>0. The property of the departure process which receives the most attention is the covariance of pairs of departure intervals, although a more thorough investigation of the probabilistic structure is undertaken for the case N = 1. A main result of the paper is that in the M/G/1 queue with N = 1, departure intervals separated by one or more intervals are independent. When G is the deterministic server, the departure stream is a renewal process. The paper concludes with an expression for the covariance of any pair of intervals in the departure process for any value of N, and numerical values for correlations in the case of a deterministic server with N = 2.
A Note on Barnard's Modification to Wald's Sequential Test for Double Dichotomies
Authors: Fryer M. J.
Published: December 2018

It is found that Barnard's modification can have the effect of making the operating characteristic of Wald's test continuous.
Properties of Some Two-Sided Sequential Tests for the Normal Distribution
Authors: Billard L.
Published: December 2018

A general procedure for sequentially comparing a null hypothesis with a two-sided alternative hypothesis concerning a normal mean is described and its properties are derived. The procedure is defined in terms of geometrical parameters which can be evaluated so as to give an optimum procedure. A particular set of specified values for these parameters permit close approximations to the O.C. function and the A.S.N. function for the Wald procedure and the Armitage procedure to be obtained.
A Note on the Admissibility of a Response Surface Design
Authors: Kupper Lawrence L.
Published: December 2018

A generalization of a result by Box and Draper (1959) on the minimization of integrated squared bias is used in conjunction with some ideas from decision theory to develop the notions of variance (V-), bias (B-) and mean square (MS-) admissibility, which have particular appeal to response surface workers. A theorem is given characterizing an important situation in which an experimental design can be said to possess one of these types of admissibility.
Multifactor Mixture Experiments
Authors: Nigam A. K.
Published: December 2018

In this paper a new class of mixture experiments which allow simultaneous consideration of proportions of several factors have been described. Practical examples have been described to elucidate these multifactor mixture experiments. Substitutions involved in a quadratic model have been derived and the procedure of obtaining block designs has been described.
Survey Design, Symmetry and Posterior Distributions
Authors: Scott Alastair , Smith T. M. F.
Published: December 2018

It is now well known that the sample design is irrelevant in a Bayesian analysis of survey data provided the units are labelled and the sampling is non-informative. However, information about the labels is often not available for the analysis, and in this situation the design will contain information about the labelling and hence affect the inferences. There is one important exception: if the prior distribution is exchangeable and we are interested only in symmetric functions of the unit values then the design is still irrelevant even without the labels.
Scientific Reporting and Personal Probabilities: Student's Hypothesis
Authors: Dickey James
Published: December 2018

It is suggested here that a scientific researcher can report his statistical data by summarizing the application of Bayes's theorem to whole ranges of prior distributions, devised to include the opinions of his readers. The Jeffreys-Savage view is taken that interesting actual prior opinions can be usefully approximated by assuming positive probability on a sharp null hypothesis (point hypothesis). Graphical tools and bounds are given for reporting the weighted likelihood ratio, or Bayes factor, the ratio of the posterior odds to the prior odds, for a vague alternative to a null value for the mean of a Normal distribution.
Dependency of Intervals between Events in Superposition Processes
Authors: Lawrance A. J.
Published: December 2018

Arbitrary event initial conditions for the superposition of a finite number of stationary point processes are derived; these are used to obtain the general joint distribution for the intervals between events following an arbitrary event. Specific consideration is given to the univariate, bivariate and trivariate distributions of successive intervals between events. Various numerical results are given for the first two serial correlations.
Multisample Tests for Multimodal and Axial Circular Populations
Authors: Mardia K. V. , Spurr B. D.
Published: December 2018

We propose an l-modal c-sample non-parametric test U on a circle based on uniform scores. The test is derived with the help of an l-modal von Mises type distribution. Critical values of the distribution are given and the asymptotic distribution of U and its parametric competitor V are obtained. For the shift-type alternatives we derive Bahadur's efficiency of the U-test relative to the V-test and consider some particular cases. We also examine the small sample powers of U and V for the von Mises type distributions. For von Mises populations, the power of the U-test is studied in relation to the power of another reasonable non-parametric test U'. We then indicate how these tests can be used for the axial case. Numerical examples are given to illustrate the use of the tests.
Bayesian and Classical Analysis of Poisson Regression
Authors: El-Sayyad G. M.
Published: December 2018

This paper is concerned with the problem of testing the existence of a trend in the means θi of Poisson distributions. It is assumed that these means are changing exponentially, that is, log θi = α + βxi. A classical method is reviewed which is used for testing the hypothesis β = 0. The exact Bayesian distribution for β is derived and a Bayesian approximation suggested which proved to be very useful. Finally, a comparison of these three methods by means of numerical examples is made.
The Measurement of Belief
Authors: Suppes Patrick
Published: December 2018

This paper criticizes some of the claims of the standard theories of subjective probability. The criticisms are especially oriented toward the structural axioms that cannot be regarded as axioms of pure rationality and the general results that yield exact measurement of subjective probabilities. Qualitative axioms for upper and lower probability are introduced to provide a theory of inexact measurement of subjective probability. Only minor modifications of de Finetti's qualitative axioms yield the desired theory. The paper concludes with a comparison of the measurement of belief to the measurement results characteristic of Euclidean geometry, and also examines briefly some possibilities for using learning models as simplified abstract processes for constructing belief.
Expectation and Dispersion Matrices for the Estimation of Distributions Associated with Discrete Reservoirs
Authors: Pegram G. G. S.
Published: December 2018

Approximate expressions are found, in the case of a particular model for the input transition matrix, for the mean and dispersion matrix of its elements and of the elements of the storage distribution for Moran and Lloyd reservoirs.
Bayesian Single Sampling Acceptance Plans for Finite Lot Sizes
Authors: Thyregod P.
Published: December 2018

We discuss a general model for acceptance/rejection decisions regarding finite populations.

The model is based on an exchangeable prior distribution of the population characteristics and additive costs. A fundamental property of the model is that the original two-decision problem regarding the characteristics of the finite population is equivalent to a classical decision problem regarding a parameter in the prior distribution of population characteristics. In the general discussion of the model we verify some simple results for the optimal sampling scheme, viz. that the expected decision cost per individual is decreasing and the optimal sample size is increasing with the population size.

A substantial part of the paper is devoted to the study of monotonely regular problems, i.e. problems admitting a sufficient statistic with a monotone likelihood-ratio. A main result of the discussion is that the optimal decision criterion is a monotone function of the sample size.

It is shown that the theory describes the standard situations in quality control with lot-by-lot sampling, where the distribution of the item characteristics belongs to an exponential family, and in an example we suggest a simple algorithm for determination of the Bayesian sampling scheme.
Confounding in Factorial Experiments
Authors: Raja J. Sundara
Published: December 2018

Various methods of construction of confounded designs and determination of the confounded effects of a confounded design are available in the literature. An alternative method based on the theory of groups is given in this paper. The problems of construction of the Principal Block (P.B.) and determination of the effects confounded from the elements of P.B. are made simple for any number of factors in 2n factorial experiments.
Bayesian Estimation Methods for Two-Way Contingency Tables
Authors: Leonard Tom
Published: December 2018

Estimation methods are proposed for the row, column and interaction effects in two-way contingency tables, the one-way table being treated as a special case. The methods are appropriate when the parameters are thought a priori to be related to each other. The posterior estimates have the practical effects of smoothing the contingency table, and are valid even if some of the cell frequencies are zero. The main case treated is where particular assumptions of exchangeability are reasonable a priori for the unknown parameters. Some possible relaxations of the exchangeability assumptions are discussed. The methods are used to provide a numerical analysis of a mobility table originally due to Karl Pearson, and to measure the association between the occupations of fathers and sons in this table.
Single Replicate Factorial Experiments in Generalized Cyclic Designs: I. Symmetrical Arrangements
Authors: John J. A. , Dean A. M.
Published: December 2018

Single replicate generalized cyclic designs are constructed from a set of generating treatments. This flexible class of designs is used to provide a general method of confounding in symmetrical factorial experiments. It is shown how the confounding scheme can be determined from an examination of the generating treatments.
Equivalence Theorems for Polynomial-Projecting Predictors
Authors: Godolphin E. J. , Harrison P. J.
Published: December 2018

Under certain conditions, the predictors of Integrated Moving Average processes, Markov Polynomial processes, Multiple Exponential Smoothing models and Exponentially Weighted Regression models are identical. In this paper, these conditions are determined and some implications and generalizations are examined.
Independent Inter-Departure Times in M/G/1/N Queues
Authors: Daley D. J. , Shanbhag D. N.
Published: December 2018

This paper extends the results of Finch (1959) and King (1971) on the stationary departure process of the queueing system M/G/1/N by establishing, among others, the existence of the systems other than M/D/1/1 and M/M/1/∞ with two successive inter-departure times independent.
The Geometry of Generalized Inverses
Authors: Kruskal William
Published: December 2018

Generalized inverses of linear transformations must satisfy at least the natural requirement that they are true inverses for appropriately restricted subspaces. There are three other characteristics that may or may not hold independently. The four properties are described in coordinate-free geometrical terms, followed by discussion of dimensionalities and other topics.
Construction of Simple Empirical Bayes Estimators
Authors: Maritz J. S. , Lwin T.
Published: December 2018

An empirical Bayes estimator which can be constructed without explicit estimation of the prior distribution is called a simple empirical Bayes estimator. Construction of such estimators for a single unknown parameter of a general continuous distribution is considered. Examples are given and their performance is examined.
Errors of Prediction for Markov Chain Models
Authors: Bartholomew D. J.
Published: December 2018

When Markov chain models are used for predicting the state distribution the forecasts are liable to error for a variety of reasons. The sources of such error are identified and methods are given for finding the contribution which each source makes to the total error of prediction. In particular it is shown that the error arising from the estimation of the parameters is likely to be of the same order of magnitude as the inherent variation arising from the stochastic assumptions of the model. The theory is first derived for a closed system and then generalized to two kinds of open system.
Estimation of Linear Functional Relationships: Approximate Distributions and Connections with Simultaneous Equations in Econometrics
Authors: Anderson T. W.
Published: December 2018

When both variables are subject to error, a straight line may be fitted by minimizing the sum of squared distances of the observed points to the line. Approximate distributions of the slope of this line and of the angle it makes to one axis are given when the errors are normal, uncorrelated, and with equal variances. These distributions and corresponding ones for the line fitted by ordinary least squares are found by making the model equivalent to a model of simultaneous equations in econometrics.
Plausibility Inference
Authors: Barndorff-Nielsen O.
Published: December 2018

A scheme for inference—plausibility theory—is proposed. It parallels likelihood theory, the analogue of the likelihood function L(ω) = p(t; ω) being the plausibility function π(ω) = p(t; ω)/supt  p(t; ω), but likelihood and plausibility throw light on different aspects of the evidence in the data t. The ordinate test for significance (which rejects for small values of the probability (density) function, of a simple hypothesis), the maximum ordinate estimate (which equals the set of ω's such that p(·; ω) has mode at t), and M-ancillarity are, to some extent in modified form, integrated in plausibility theory. Among the subthemes, particular attention is given to relevant properties of universality (in its origin a preconcept for M-ancillarity), to exponential models, and, naturally, to similarities and contrasts between plausibility inference and likelihood inference. The logic connected with the ordinate test is also attended to.
Approximating the Lower Tail-Areas of the Kolmogorov–Smirnov One-Sample Statistic
Authors: Pelz Wolfgang , Good I. J.
Published: December 2018

Various methods for calculating the Kolmogorov–Smirnov one-sample statistic have been developed in the literature. A transformation of an approximation method is here derived and some of its properties discussed. The main value of the new formulae is to obtain more convenient approximations in the lower tail than have been possible using other methods. The relationships between various methods are given, as well as recommendations for each method of a usable range of the independent variable.
Some Simultaneous Confidence Bounds in Normal Models with Restricted Alternatives
Authors: Marcus Ruth , Peritz Eric
Published: December 2018

Simultaneous lower confidence bounds are presented for certain restricted normal models. Applications are given for certain linear contrasts of the means in one-way anova with ordered alternatives and for linear functions of the means in one-way anova with non-negative means.