bluuebunny/crossref_metadata_2025 huggingface dataset:
A Minimax Procedure for Choosing between Two Populations Using Sequential Sampling
Authors: Maurice Rita J.
Published: December 2018

When a choice has to be made in favour of one of two populations the cost of sampling (experimenting) in order to obtain information on which to base the decision must be balanced against the cost of making the wrong choice. Wald’s (1950) minimax procedure is used to determine a sequential method of sampling when a decision is to be made between two normal populations on the basis of their sample means. It is assumed that the cost of making a wrong choice is given by the difference of the population means multiplied by a factor representing the use made of the chosen population. The population variances are taken as known and equal. It is found that this determines a sequential procedure defined by the ratio of the difference it is desired to detect to a function of the probability with which it is desired to detect that difference, therefore corresponding to a range of tests in which these constants are fixed for the statistician by the user of the results. The performance of this procedure is compared with that for a fixed sample size and found to result in smaller losses over all values of the difference between the populations. A comparison is also made with the double sample procedure of Grundy et al. (1956) on which it is also generally an improvement.
The Lindisfarne Scribes' Problem
Authors: Silvey S. D.
Published: December 2018

The Lindisfarne scribes' problem is one of estimating, from observations on an ordered set of random variables, the number of changes which occur in the distribution functions of these random variables as they are considered in order. This is a multiple decision situation and it is suggested that, in the absence of a loss function, the principle of controlling the probability of over-estimating the number of changes, which is a natural extension of the principle underlying many currently applied tests of significance, might be used. An estimation procedure, based on this principle, is suggested for the Lindisfarne scribes' problem, and some of its implications are discussed.
The Asymptotic Powers of Certain Tests of Goodness of Fit for Time Series
Authors: Hannan E. J.
Published: December 2018

The distances, upon which the asymptotic powers of certain goodness of fit tests for stationary time series depend, are expressed in terms of the spectral density functions on the null and alternative hypotheses.
A Difference Equation Technique Applied to the Simple Queue
Authors: Conolly B. W.
Published: December 2018

The difference equations satisfied by the Laplace transforms of the state probabilities at finite time are solved by a direct method in the case of a simple queue. This is an alternative to Bailey’s (1954) procedure, which appears to economize in algebra.
Dilution Series: A Statistical Test of Technique
Authors: Stevens W. L.
Published: December 2018

When only the presence or absence of a micro-organism can be ascertained, density can be determined by preparing a dilution series in geometrical progression, setting aside the same number of plates at each level and observing fertile and sterile plates. Sometimes, series of observations appear which suggest that some failure in technique has occurred. An example, in a ten-fold dilution series, with five plates at each level, is given by the following number of fertile plates:

… 5 5 4 1 5 5 5 4 2 0 0 …

To provide an objective criterion to measure the significance for a possible deficiency of technique, the writer proposes the range, defined as the number number of levels, counted inclusively, from the first where not all plates are fertile to the last where not all are sterile (in the above example, 7). The distribution of the range is found, and tables provided for making a test of significance.
Equally Correlated Variates and the Multinormal Integral
Authors: Stuart Alan
Published: December 2018

An  n-dimensional distribution with all correlations equal to p, called an equally correlated distribution, can very simply be constructed from a set of n + 1 one-dimensional variates; these variates are uncorrelated if p is non-negative, and n of them are uncorrelated if p is negative. From this, a general relation is deduced between any problem concerning an equally correlated distribution and an equivalent problem for uncorrelated variates. Applications are (i) to a t test due to Walsh (1947) and Halperin (1951); (ii) to Moran’s (1956) result for the probability that all the variates of an equally positively correlated normal distribution are positive and to a generalization of this; (iii) to the approximation of sampling distributions under randomization to the results of standard normal theory.
The Process Curve and the Equivalent Mixed Binomial with Two Components
Authors: Vagholkar M. K.
Published: December 2018

A method is given and illustrated for obtaining a two-ordinate process curve equivalent, for acceptance sampling problems, to a given process curve.
On Some Problems of Machine Interference
Authors: Nasr Saad K.
Published: December 2018

The problem of interference arises when a team of operatives is attending a set of machines, these being assumed to have a certain servicing factor. In this paper an estimator of the servicing factor and a simple expression for the machine efficiency are introduced. A loss function is also given, with a method for the determination of an appropriate number of operatives to be applied for the attendance of the machines.
Bandwidth and Resolvability in Statistical Spectral Analysis
Authors: Lomnicki Z. A. , Zaremba S. K.
Published: December 2018

The purpose of this paper is to refute Grenander’s latest objections to the authors' work on statistical spectral analysis, and to clarify the distinction between resolvability, bandwidth, and bias.
On a Discriminatory Problem Connected with the Works of Plato
Authors: Cox D. R. , Brandwood L.
Published: December 2018

A form of discriminant analysis for qualitative data is developed and used to place in order some of the works of Plato.
The Efficiency of Automatic Winding Machines with Constant Patrolling Time
Authors: Howie A. J. , Shenton L. R.
Published: December 2018

The problem is that of maximizing the average efficiency of a machine which (i) stops when its total running time has reached a fixed value a, (ii) has accidental stops, and (iii) is patrolled for servicing (i.e. re-starting if necessary) at a fixed interval p which may be less than a, efficiency being defined as the ratio of the running time to the total time under consideration. The probability distribution of the number of patrols required to exhaust the total running time of the machine is derived in the case where the accidental stops have uniform probability, and is found to be expressible in terms of Laguerre polynomials. The properties of the distribution are illustrated by an example. its asymptotic normality as a increases is established. The properties of the average efficiency are discussed.
Confidence Regions in Non-Linear Estimation
Authors: Beale E. M. L.
Published: December 2018

The statistical properties of the approximate confidence regions for nonlinear estimation based on the likelihood ratio criterion are considered, assuming normal variation of errors. It is suggested that these approximate confidence regions are preferable to any known exact confidence regions for this problem, provided that the model is only moderately non-linear.

A numerical measure of the non-linearity of a model (as a function of the experimental design and the parameter values) is proposed. This can be used to indicate: (a)the accuracy of these approximate confidence regions, and a simple method of modifying the regions to avoid exaggerating the significance of the experimental data in doubtful cases; and(b)the effectiveness of a non-linear transformation of parameters as a device for making the model approximately linear.

Methods of computing the non-linearity so defined, either empirically or theoretically, are indicated.
Discrete Stochastic Processes in Population Genetics
Authors: Bodmer W. F.
Published: December 2018

The application to population genetic models of the approximate treatment of a non-linear discrete stochastic process by linearizing near an equilibrium point is discussed. The value of the dominant characteristic root for approximate linear changes near an initial equilibrium can be used to estimate the chances of survival of newly occurring genotypes. These methods are illustrated on the classic population genetic model for a balanced polymorphism and also on a model for a simple incompatibility system. A method is developed for the complete stochastic treatment of a linear process. This makes it possible to fit observed changes to those expected from theoretical models when it is assumed that the changes are near an equilibrium point. It is also possible to estimate the chance of losing a gene segregating with a given selective advantage in a small population of known size and in certain cases to obtain estimates of the selective advantage when a certain proportion of fixations has been observed. A general treatment of a complex model for the evolution of homostyly in primroses using these methods is given. Numerical methods, using an electronic computer, are applied to both deterministic and stochastic study of this model.
Estimation of Parameters of a Multivariate Normal Population from Truncated and Censored Samples
Authors: Singh Naunihal
Published: December 2018

This paper is concerned with the problem of estimation of means and variances of a k-variate normal distribution from a sample doubly truncated or censored on s-variates (s ≤ k), when variates are uncorrelated. Maximum likelihood equations are derived which, with the help of normal probability tables, can readily be solved by iteration. The information matrix is given from which the asymptotic variances and covariances of the estimates can be determined. to illustrate the practical application of these results a numerical example is added.
The Wilcoxon Test and Non-Null Hypotheses
Authors: Wetherill G. B.
Published: December 2018

The two-sample Wilcoxon test provides an exact test of location when the two parent populations are identical in shape. The research recorded in this paper is an investigation into the behaviour of the test, and a comparison with the behaviour of the t-test, when the two distributions differ slightly in shape.
A Simple Method of Trend Construction
Authors: Leser C. E. V.
Published: December 2018

The principle adopted here in the construction of a trend for a time series consists in minimizing a linear combination of two sums of squares, of which one refers to the second differences of the trend values, the other to the deviations of the observations from the trend values. Properties of the general solution are deduced, and the solution is explicitly obtained for up to 7 observations.

In the special case in which the sum of the two sums of squares is minimized, the exact solution is derived for up to 15 observations. An approximation formula, suitable for practical use when there are 8 or more observations, is also given. The method is illustrated by examples, in which it is applied to an artificially constructed and to an actual time series.
An Unbiased Estimator for Powers of the Arithmetic Mean
Authors: Glasser Gerald J.
Published: December 2018

An unbiased estimator for integral powers of the arithmetic mean, when sampling with replacement, is given, together with a formula for its variance and an unbiased estimator for this variance. Similar results are given for functions which can be expressed as a series in powers of the mean.
A Test of Homogeneity of Means Under Restricted Alternatives
Authors: Bartholomew D. J.
Published: December 2018

Suppose that observations x1, x2, …, xk have been obtained from normal populations with unknown means μ1, μ2, …, μk and known standard deviations σ1, σ2, …, σk. The paper discusses tests of the hypothesis μ  1, μ  2, … = μk when prior information is available about the μ‘s under the alternative hypothesis. A generalized form of the χ  2-test, denoted by x2¯, has been developed for use when this information takes the form of order restrictions between some or all of the μ‘s. The distribution is shown to depend on that of χ  2 and certain probabilities which have been determined for many sets of order restrictions. The power function has been obtained in certain cases and used to compare the x2¯-test with χ  2, which assumes no prior information. This shows that the utilization of order information often leads to a considerable gain in power. Power comparisons with other versions of x2¯ and with tests based on scores suggest that no worthwhile improvement over x2¯ is possible. The paper is an extension and generalization of work contained in two earlier papers (Bartholomew, 1959a, b).
Closed Queueing Systems: A Generalization of the Machine Interference Model
Authors: Benson F. , Gregory G.
Published: December 2018

A closed loop of queues in series is considered in which the time taken by any customer to travel between consecutive service gates is a random variable. The case of a single server at each gate is analysed, where in addition to the customers from the closed loop, each server may also have customers arriving from sources outside the system and, after completion of service, departing to destinations again outside the system. Limiting cases are examined, firstly where the number of customers in the closed loop becomes large, and secondly where, in addition to this condition, the average transit time between one pair of consecutive service gates also tends to infinity.
The Randomization Bases of the Problem of the Amalgamation of Weighted Means
Authors: Barton D. E. , Mallows C. L.
Published: December 2018

A generalization is given of a theorem of Miles on the probability distribution of the number of resultant means arising from a process of amalgamation of a set of weighted values, under an hypothesis of randomization. Two proofs are given, one inductive and one combinatorial. These are based on a geometrical representation. The theorem is used to provide a non-parametric test for trend. The connection of this probability distribution with that of the number of records (in the sense of Foster and Stuart) is explored and an explicit formula for the variance of their round trip test is obtained. Two tests, based on the number of records and on the number of resultant means, are applied to two sets of data. Three standard tests are also applied for comparison.
A Method of Maximum-Likelihood Estimation
Authors: Richards F. S. G.
Published: December 2018

Knowledge of one or more of the parameters in a maximum-likelihood problem sometimes results in a very considerable simplification, the analysis becoming almost trivial. A method of obtaining maximum-likelihood estimates and their asymptotic covariance matrix is given which makes it possible to capitalize on simplifications of this sort when they arise. Application is mainly in the field of regression, but other examples are mentioned. A numerical example of exponential regression is given.
A Waiting Line with Interrupted Service, Including Priorities
Authors: Gaver D. P.
Published: December 2018

A single-server system with stationary compound Poisson input and general independent service times, the latter being subject to random interruptions of independently but otherwise arbitrarily distributed durations, is studied. For a variety of service-interruption interactions (including the preemptive-repeat) the distributions of busy period duration, of queue length, and of waiting time are characterized by transforms and by moments. Applications are made to priority scheduling problems.
On the Distribution of Certain Quadratic Forms in Normal Variates
Authors: Bhat B. R.
Published: December 2018

This paper is concerned with the characterization of the variance covariance matrix for a normal family in which certain quadratic forms are independent and follow cχ  2 distributions. It is also proved that the necessary and sufficient condition for any quadratic or linear form to be distributed independently of a sum of a finite number of non-negative definite quadratic forms is that it is distributed independently of each one of them.
A Note on Interactions in Contingency Tables
Authors: Plackett R. L.
Published: December 2018

Parametric definitions of zero second-order interaction in a threefold contingency table are discussed, and some related tests are considered. A general test is constructed by extending Woolf’s criterion for the 2 × 2 × t table.
Estimators for the Product of Arithmetic Means
Authors: Glasser Gerald J.
Published: December 2018

Symmetric functions which provide unbiased estimators for the product of m population arithmetic means are considered for two sampling procedures: (a) m independently selected simple random samples, and (b) a simple random sample of multivariate observations. Formulae for the variances of these estimators and for estimated variances are derived.
Confidence Sets for the Mean of a Multivariate Normal Distribution
Authors: Stein C. M.
Published: December 2018

An attempt is made to determine confidence sets for the mean of a multivariate normal distribution with known covariance matrix that take advantage of the fact that the sample mean is not the best estimate when the loss is a non-singular quadratic function of the error vector. Only the case of high dimension is considered. The geometrical size and shape of the confidence sets, the probability of covering false values, and the relation to posterior probabilities are studied, unfortunately somewhat incompletely.
Some Statistical Aspects of Adaptive Optimization and Control
Authors: Box G. E. P. , Jenkins G. M.
Published: December 2018

It is often necessary to adjust some variable X, such as the concentration of consecutive batches of a product, to keep X close to a specified target value. A second more complicated problem occurs when the independent variables X in a response function η(X) are to be adjusted so that the derivatives ∂η/∂X are kept close to a target value zero, thus maximizing or minimizing and the paper is devoted mainly to the estimation from past data of the “best” adjustments to be applied in the first problem.
On the Lengths of Intervals in a Stationary Point Process
Authors: McFadden J. A.
Published: December 2018

The paper is concerned with the lengths of intervals in a stationary point process. Relations are given between the various probability functions, and moments are considered. Two different random variables are introduced for the lengths of intervals, according to whether the measurement is made from an arbitrary event or beginning at an arbitrary time, and their properties are compared. In particular, new properties are derived for the correlation coefficients between the lengths of successive intervals. Examples are given. A theorem is proved, giving conditions under which two independent stationary point processes with independent intervals may be superposed, giving a new point process which also has independent intervals. Mention is made of the application to the theory of binary random processes and to the zeros of a Gaussian process.
Problems of Selection with Restrictions
Authors: Rao C. Radhakrishna
Published: December 2018

The paper considers a practically important generalization of the theory of regression. A linear function of a set of variables x  l, …, xp, called predictor variables, is constructed so as to maximize its correlation with a criterion variable y  1 subject to the condition that its correlations with other criterion variables y  2,…,yq are non-negative. It is suggested that a linear function so determined is useful when selection of individuals is done on the basis of x  1,…,xp to achieve the maximum possible progress in the mean of y  1, while ensuring that no deterioration takes place in the mean values of y  2,…,yq in the selected group, compared with the original group of individuals from which selection is made.
Approximate Confidence Limits for the Relative Risk
Authors: Gart John J.
Published: December 2018

The relative risk or odds ratio has been proposed as one alternative to the difference in proportions in measuring the degree of association in two by two tables. This paper proposes two new methods for finding confidence limits on the relative risk, one appropriate for large numbers, and a second for small numbers. The first is related to the usual chi-square test and the second to an approximation to the Fisher-Irwin test. The methods are compared to those previously proposed by Cornfield (1956), Cox (1958), and Woolf (1954).
The Use of Control Observations as an Alternative to Incomplete Block Designs
Authors: Atiqullah M. , Cox D. R.
Published: December 2018

The use of a control treatment as an alternative to incomplete block designs is considered, two possible methods of analysis being proposed. Theoretical efficiency factors are evaluated. These confirm conclusions of Yates (1936) that the efficiency is lower with a control than with an appropriate incomplete block design. The difference is not great for the larger values of k, the number of units per block, and the greater flexibility and simplicity of analysis of the methods with a control may lead to their being preferred, especially when a control treatment is necessary for other reasons, or when no simple incomplete block design is available. Extensions to the two-way elimination of error are considered.
Confidence Interval Estimation in Non-Linear Regression
Authors: Halperin Max
Published: December 2018

In a recent paper, E. J. Williams presents a procedure for obtaining what are described as fiducial intervals and regions for the non-linear parameters in non-linear regression. The linear parameters are not included in the regions which are developed and the class of non-linear regression functions for which his results hold is not clearly delineated. In this paper, it is shown how some results from standard regression theory can be used to discuss these points; the discussion also implies that the intervals and regions described by Williams are confidence intervals and regions.
Multivariate Linear Hypothesis with Linear Restrictions
Authors: Kabe D. G.
Published: December 2018

In this paper we consider a normal multivariate linear regression model and give a criterion for the simultaneous testing of a set of linear estimable parametric functions. We also derive the distribution of the criterion and obtain some allied results.
Treatment of the Non-Equilibrium Theory of Simple Queues by Means of Cumulative Probabilities
Authors: Sack R. A.
Published: December 2018

The treatment of simple single-server queues with random arrivals and exponentially distributed serving times is facilitated by the use of cumulative sums of which the lowest order represents the probability that there are at least a given number of customers in the queue. For constant arrival and departure rates λ and μ the deviations from the steady-state conditions are essentially symmetric in λ and μ; they can be expressed as integrals over modified Bessel functions which are slightly simpler than equivalent expressions given by Bailey. Asymptotically these terms decay approximately exponentially with an effective rate κ=(λ−μ)2.
Theory of Cyclic Rotation Experiments
Authors: Patterson H. D.
Published: December 2018

The paper reviews some of the statistical problems arising in the design and analysis of long-term cyclic experiments comparing different crop rotations. Three types of design are distinguished and their properties considered. These are basic designs with all phases of the rotations in each block, reduced designs with mutually exclusive groups of phases kept in separate blocks and phase-confounded designs in which some contrasts between test crops are partially confounded with block differences.

Methods of analysing the yields of test crops by estimating the mean effects of the rotations over the years and regressions on seasonal and time variates are discussed theoretically for replicated experiments of basic or reduced design. The analysis is complicated by correlations between yield values recurring on the same plots and by lack of homogeneity in residual year-to-year variations in rotation effects. The main topics considered are (1) the estimation of errors, (2) the losses of information due to using unweighted means and regressions ignoring the correlations, (3) methods for recovering this information.
Prediction by Progressive Correction
Authors: Dobben de Bruyn C. S.
Published: December 2018

A new kind of non-linear predictor is introduced, called a progressive corrector. The predictor is closely related to the exponentially weighted moving average, which can be regarded as a degenerate case of a progressive corrector. An attempt is made to approximate the mean square error of prediction. Mean square error is criticized as a criterion for the optimality of a predictor which is being used as part of an inventory control system.
Two Papers on the Comparison of Bayesian and Frequentist Approaches to Statistical Problems of Prediction: Bayesian Tolerance Regions
Authors: Aitchison J.
Published: December 2018

In the theory of statistical tolerance regions, as usually presented in frequentist terms, there are inherent difficulties of formulation, development and interpretation. The present paper re-examines the basic problem from a Bayesian point of view and suggests that such an approach provides a set of widely applicable, mathematically tractable tools, often more tailored to the requirements of users than the corresponding frequentist tools. For the one-dimensional case, Bayesian intervals are quoted for a number of standard distributions and prior densities, and the customary feature of a Bayesian analysis—that special prior densities give rise to standard frequentist results—is briefly demonstrated. A problem which seems to be of greater practical significance, namely the selection of an optimum tolerance region from a set of possible tolerance regions, is also investigated and the overwhelming advantages of the Bayesian approach are indicated.
On the Bayesian Estimation of Multivariate Regression
Authors: Tiao George C. , Zellner Arnold
Published: December 2018

In this paper, we use a Bayesian approach to analyse sets of regression equations with correlated error terms. In the case that the matrix of “independent variables” is the same for all equations, our model reduces to the traditional multivariate regression model. For this case, the marginal posterior distribution of the regression coefficient vector for any equation is shown to be of the multivariate-t form. Further, the variances and covariances of the error terms have an “inverted” Wishart distribution a posteriori. Some properties of this distribution are given. Finally, the joint posterior distribution of the regression coefficients in the general model is derived and discussed.
A Note on the Solution of Dam Equations
Authors: Lloyd E. H. , Odoom S.
Published: December 2018

The object of this paper is to provide an algorithm for solving the equations giving the equilibrium distribution of reservoir levels, both for Moran's formulation in terms of independent inflows and for Lloyd's formulation involving Markovian inflows. In the case of independent inflows, the equilibrium distribution may be expressed linearly in terms of powers of the roots of a certain polynomial. When the inputs form a Markov chain, the solution may be similarly expressed in terms of the latent roots of a certain matrix, or (more conveniently for computation) directly in terms of powers of that matrix. The method is illustrated by simple examples.
A Model for Rainfall Occurrence
Authors: Green J. R.
Published: December 2018

A probability model for the occurrence of wet and dry days at Tel Aviv is proposed as an alternative to an earlier one given by Gabriel and Neumann. The new model, which assumes exponential distributions for the lengths of spells of continuous dryness or continuous rain, is found to fit rainfall data for Tel Aviv well, and to represent certain conditional probabilities more satisfactorily than that of Gabriel and Neumann.
On an Identity for the Variance of a Ratio of Two Random Variables
Authors: Koop J. C.
Published: December 2018

A new identity showing the relationship between the variance of the ratio of two random variables and the variance of the random variable in the numerator of the ratio is presented. For any sample design the identity offers a basis for exploring the conditions under which ratio estimators have smaller variance than linear estimators.
On Circular Functional Relationships
Authors: Chan Nai Ng
Published: December 2018

Problems concerning the consistency of estimators of the parameters of a circular functional relationship are considered, and a practical procedure for obtaining estimates is provided.
Probability Bounds for a Union of Hyperspherical Cones
Authors: Choi Keewhan
Published: December 2018

In this paper are given two methods of obtaining bounds on the probability content of a union of hyperspherical cones in Euclidean N-space when the underlying distribution is normal. The results of a small numerical study are tabulated to indicate the range of applicability of the methods. These methods can be used to obtain bounds on the overall significance level and power-like probabilities in the analysis of variance when many treatment sums of squares are tested against the same error sum of squares.
Formulae to Improve Wald's Approximation for Some Properties of Sequential Tests
Authors: Tallis G. M. , Vagholkar M. K.
Published: December 2018

Wald's fundamental identity for a sequential probability ratio test is derived in a rather different way and, with some assumptions, it is used to derive formulae to improve Wald's approximations for the operating characteristic and the average sample number functions.
Association between Random Variables and the Dispersion of a Radon–Nikodym Derivative
Authors: Ali S. M. , Silvey S. D.
Published: December 2018

In a previous paper by one of the authors (Silvey, 1964) it was suggested that the Radon–Nikodym derivative of the joint distribution of two, not necessarily real-valued, random variables with respect to the product of their marginal distributions provides the analytic key for discussing association between the random variables. In this paper we shall develop this point and lend some support to the suggestion by proving that in the case of normal random vectors the distribution of this derivative, as determined by the product of the marginal distributions, becomes more widely spread out as association between the vectors increases. More precisely we shall show that the expected value of any continuous convex function of the derivative is a non-decreasing function of each of the canonical correlation coefficients between the two vectors.
The Detection of Partial Association, II: The General Case
Authors: Birch M. W.
Published: December 2018

Three criteria are derived for testing null hypotheses of conditional independence of two random variables. The first of these criteria is appropriate when both are qualitative, the second when both are quantitative, and the third when one is qualitative and one quantitative. Each of these criteria may be regarded as a generalization of the criterion for two dichotomous random variables given in Part I.
A Branching Process Allowing Immigration
Authors: Heathcote C. R.
Published: December 2018

It is well known that in the absence of immigration, a population of like particles developing under the usual laws for branching processes either increases unboundedly with time or becomes extinct (Feller, 1957, Chapter 12; Harris, 1963, Chapter 1). If migration into the system is permitted, then it is clear that under certain conditions a proper stationary distribution for population size will exist. That this is the case has been shown by Bartlett (1956, Section 3.41) for processes in continuous time, and the present note is concerned with the same problem but considering discrete generations. Our result for the generating function of the stationary distribution of population size reduces to a form analogous to Bartlett's result when the immigration distribution is Poisson.
Transformations and Sufficiency
Authors: Sprott D. A.
Published: December 2018

Barnard (1963) gave a definition of the sufficiency of T  2 for θ  2 in the absence of knowledge of θ  1 using groups of transformations. A possible generalization of this is implicit in the method of obtaining fiducial distributions by sets of transformations described by Sprott (1963). The purpose of this paper is to examine sufficiency and transformations in more detail in the special cases relevant for fiducial distributions, and to examine the nature of possible groups of transformations.
Likelihood-Ratio and Confidence-Region Tests
Authors: Aitchison J.
Published: December 2018

A previous paper (Aitchison, 1964) proposed a method of confidence-region testing for multiple-hypothesis situations and developed the theory for linear univariate situations and for the asymptotic parametric case. The present paper shows that the results for these situations are in fact special cases of a basic relation between confidence-region tests and likelihood-ratio tests. This relation allows the easy extension of confidence-region testing to multivariate analysis.
Analytical Surveys with Cluster Sampling
Authors: Sedransk J.
Published: December 2018

This paper concerns the designing of certain analytical surveys (utilizing cluster sampling). The general concepts and problems peculiar to this type of study are discussed. The question of the relevant population for inference purposes is treated, and this is reflected in the model considered. Illustrations are given to clarify the situations where this type of model may be applicable. In general, the objective in planning such a survey is to select the sample so that maximum precision for the comparisons to be made is obtained for a specified total cost. Sample size allocations are carried out for several specifications of survey goals—these specifications being related to the type of comparisons to be made.
Finite Stopping Time and Finite Expected Stopping Time
Authors: Savage I. R. , Savage L. J.
Published: December 2018

Generalized sequential probability-ratio procedures are defined for dependent and non-identically distributed random variables. For these procedures, conditions are found implying that the stopping time is finite with probability 1 and the expected stopping time is finite. These results are applied to rank order problems.
Moments of a Serial Correlation Coefficient
Authors: Shenton L. R. , Johnson Whitney L.
Published: December 2018

For a discrete first-order auto-regressive scheme (with initial value zero) xt = αxt-1 + ϵt, expressions have been found for the first four moments of α̂=∑1nxtxt−1/∑1n−1xt2. The expressions are of two kinds : (i) series in ascending powers of α2, (ii) asymptotic series in descending powers of n. A table of values of μ1‱,σ(α̂),β1(α̂),β2(α̂) is included. For the special case α2 = 1 the dominant terms in the first four moments are stated.
Expected-Cover and Linear-Utility Tolerance Intervals
Authors: Aitchison J.
Published: December 2018

Expected-cover tolerance intervals are commonly used as a means of statistical prediction. In their construction the effectiveness of such intervals in relation to a future observation y is assessed only in terms of “success” if y falls in the interval and of “failure” otherwise; no account is taken in the assessment of how far inside or outside the interval y happens to fall. The present paper considers the construction of intervals which do take account of this factor in a linear way. From a Bayesian viewpoint it is found that expected-cover and linear-utility intervals can be regarded as equivalent through a simple relation between the expected cover and the relative cost ratio. For the frequentist approach it is first shown that linear-utility intervals can be simply constructed for the normal and gamma distributions. Comparison of these with expected-cover intervals shows that, while there is not complete identity, there is an equivalence in a “large-sample” sense.
The Analysis of Variance of Designs with Many Non-Orthogonal Classifications
Authors: Rees David H.
Published: December 2018

A method of estimating means and standard errors is given for designs with many classifications, whether mutually orthogonal or not. The three-classification case is examined in detail, and the computational scheme exhibited.
Classifying Youden Rectangles
Authors: Preece D. A.
Published: December 2018

Youden rectangles (Youden square designs) of various sizes are classified. The classification procedure is similar to that used by Norton (1939) for 7 × 7 Latin squares. For example, 7 × 4 Youden rectangles are grouped into six “species”, resembling the 147 species of 7 × 7 Latin squares, and the six species into three “families”. The classification led to the discovery of a method for generating balanced 7 × 3 and 11 × 5 row-and-column designs for two non-interacting sets of treatments, one set containing seven or eleven treatments, the other set containing three or five. The derivation of these designs is given. The existence of such designs had been conjectured by Clarke (1963).
The Chi-Square Test for Heterogeneity of Proportions, after Adjustment for Stratification
Authors: Armitage P.
Published: December 2018

In testing differences between proportions or rates of occurrence of events in different categories, expected numbers may be calculated separately within certain strata and subsequently summed over strata. For the Poisson case an approximate X2-statistic is derived as a simple modification of the usual test statistic. Other approaches and situations are discussed more briefly.
The Ergodic Queue Length Distribution for Queueing Systems with Finite Capacity
Authors: Keilson J.
Published: December 2018

Let M(t) be the queue length process for an M/G/1 queue with finite queue capacity L. A simple expression is given for the ergodic queue length distribution PL(m) = p{M(∞) = m}, in terms of certain Green's function elements for a Markov process Mk imbedded in M(t). The ergodic distribution for the imbedded process and that for the original process are shown to differ in the presence of finite queue capacity. The ergodic queue length distribution for G/M/1 with finite queue capacity is also exhibited.
On the Correlation Structure of the Departure Process of the M/Eλ/1 Queue
Authors: Jenkins J. H.
Published: December 2018

The joint distribution in the stationary case of four random variables is studied in this paper. Two of these are discrete and arise as the numbers of customers left behind in the queue by two successive departing customers in the imbedded Markov chain analysis. The other two are continuous, being the time intervals between three successive departures. The marginal joint distribution of the latter two random variables is found; in particular, the autocorrelation of lag 1 for intervals in the departure process in a stationary system is evaluated. By extending the analysis to the study of three successive departure intervals the autocorrelation of lag 2 for the process is evaluated. Burke (1956) and Finch (1959) proved that with Poisson arrivals, and service times independent for a single-server queue in equilibrium, the departure intervals are independent if and only if service time is exponentially distributed. This paper gives a measure of the dependence in the important case when the service times are special Erlangian.
Cyclic Incomplete Block Designs
Authors: John J. A.
Published: December 2018

It is shown that cyclic incomplete block designs for a given number of treatments (t) may be effectively reduced in number in view of equivalences which exist among designs of the same size. Designs are enumerated for 5 ⩽ t ⩽ 15.
On a Generalized Queueing System with Poisson Arrivals
Authors: Shanbhag D. N.
Published: December 2018

This note shows that an inductive method used by Prabhu (1960) to obtain the distribution of the busy period of the queueing system M/G/1 may be used to obtain the busy period distributions for the queueing system M1M2, …, M1/G  1, G2, …, Gr/1 first derived using a somewhat longer method by Welch (1965).
A Note on the Size of Epidemics and the Number of People Hearing a Rumour
Authors: Cane Violet R.
Published: December 2018

Models for the spread of an epidemic and of a rumour are compared; their deterministic forms are shown to be similar. The variance of the number unaffected is found approximately by using exchangeable variables.
Hypothesis Testing When the Sample Size is Treated as a Random Variable
Authors: Bartholomew D. J.
Published: December 2018

In the Neyman-Pearson theory of hypothesis testing it is customary to calculate significance levels and power functions on the assumption that the sample size is fixed. The main purpose of this paper is to propagate the view that the restriction of the theory to reference sets in which the sample size is constant is neither necessary nor desirable. The paper falls into two parts. In the first we consider the test of a simple null hypothesis versus a simple alternative. In this case it is pointed out that the optimum test requires the rejection of the null hypothesis if the likelihood ratio is less than a constant, K, which is independent of n, the sample size. In this respect it is identical with Bayesian and likelihood methods of dealing with the same problem. Two methods for determining a suitable value for K in the absence of knowledge about the distribution of sample size are proposed. These ideas are extended, in the second part of the paper, to the case of a simple null hypothesis and a composite alternative. Since no uniformly most powerful test exists, a variety of tests suggested by Bayesian and likelihood considerations are compared from a frequentist point of view. This work is exploratory, being based on numerical calculations, but provides a partial justification for treating sample size as if it were fixed.
Asymptotic Properties of Bayesian Single Sampling Plans
Authors: Hald A.
Published: December 2018

The model to be considered is based on a differentiable prior distribution, a general loss function, an operating characteristic written as an Edgeworth expansion, and sampling costs expressible as a polynomial in the sample size. Asymptotic expressions for the Bayesian single-sampling plan are derived and it is shown how previously known results may be found as special cases of the present one. The efficiency of non-optimum plans is discussed.
A Simple Procedure for Determining Upper and Lower Limits for the Average Sample Run Length of a Cumulative Sum Scheme
Authors: Kemp Kenneth W.
Published: December 2018

The method gives very simple expressions from which the properties of cumulative sum charts can be obtained for practical situations where only upper and lower bounds for the average sample run length are required.
Models of the Second Kind in Regression Analysis
Authors: Fisk P. R.
Published: December 2018

Regression models of the first and second kind are distinguished by the absence or presence, respectively, of variable regression coefficients. A characteristic of a model of the second kind is its heteroscedasticity. The problems of estimation in a general regression model of the second kind are discussed briefly. A more detailed examination is given of the properties of various estimators of the coefficients in the overall regression and scedastic curves for the linear model of the second kind. Models appropriate to replicated and unreplicated data are treated separately.
The Likelihood and Invariance Principles
Authors: Hartigan J. A.
Published: December 2018

The likelihood and invariance principles, when combined, imply the probability principle, which allows only very restricted inferences about a parameter, based on an observation. It is suggested, in explanation, that the likelihood and invariance principles are valid under two different viewpoints of the model, and that these viewpoints are incompatible.
On the Prediction of Non-Stationary Processes
Authors: Abdrabbo N. A. , Priestley M. B.
Published: December 2018

We consider the problem of linear least-squares prediction for a class of non-stationary processes which possess “evolutionary spectral representations”. It is shown that, under certain conditions, such processes admit moving-average representations in terms of time-dependent coefficients. This feature enables us to develop a close analogue of the Wiener–Kolmogorov approach to the corresponding problem for stationary processes.
Distribution-Free Sufficiency in Sampling Finite Populations
Authors: Joshi V. M.
Published: December 2018

The concept of distribution-free sufficiency, (DF-sufficiency for short) for estimation in survey sampling, which has been introduced by Godambe (1966), is investigated. Uncorrelatedness in Godambe's definition is shown to be equivalent to statistical independence, as is claimed on the author's behalf in the appendix to Godambe's paper. But the second result claimed there, that every DF-sufficient estimate must necessarily be of the form a(s).∑i∈sxi+b(s) is false, as the complete class of DF-sufficient estimates consists of estimates which involve all the variates represented in the sample. This second result, however, remains valid with an alternative definition of DF-sufficiency. A second alternative definition is also discussed.
On Some Tests for Several Linear Relations
Authors: Basu A. P.
Published: December 2018

In this paper we consider several tests for the parameters of k linear relations among the p variables ξ1, ξ2 ‥, ξp where the ξ's are not observable and p > k. This extends the work of Villegas (1964) who considered the above problem for the special case k = 1. The applicability of the above model in an econometric problem is also considered.
Least-Squares Fitting of a Polynomial Constrained to be Either Non-Negative, Non-Decreasing or Convex
Authors: Hudson Derek J.
Published: December 2018

An algorithm is provided to fit the (unique) least-squares polynomial Y(x) to n data points (x, y) subject to a constraint that it be either non-negative, non-positive, non-decreasing, non-increasing, convex or concave at all points x in a closed interval I. Some generalization to a general linear model can be handled, and certain alternative constraints can also be dealt with.

The problem is close to the classical problem of Quadratic Programming (Q.P.), and some of the known results in Q.P. may be used in fitting the curve.

The algorithm terminates in at most three steps. The number of steps depends on the degree of the polynomial Y(x) and the data. When no more than two steps are required, the algorithm is very easy to carry out but when three steps are necessary, it becomes more difficult.

A useful statistical application is the fitting of a polynomial to growth data. In that case, we might like the fitted curve to be non-decreasing over the range of interest.
Filtering Non-Stationary Signals
Authors: Abdrabbo N. A. , Priestley M. B.
Published: December 2018

We observe a record consisting of a “signal” plus “noise” up to the time instant t, and wish to extract the value of the signal at the time instant (t + m) by means of a linear filter. Here, m may be positive (corresponding to “prediction”) or negative (corresponding to “smoothing”). In this paper we consider the case where both the signal and noise processes are non-stationary and possess evolutionary spectral representations. Using this approach we obtain a close analogue of the Wiener–Kolmogorov treatment of the stationary case, and show that the optimum filter is uniquely determined by the form of the evolutionary spectra.