bluuebunny/crossref_metadata_2025 huggingface dataset:
A Contribution to the “Travelling-Salesman” Problem
Authors: Morton G. , Land A. H.
Published: December 2018

This paper is concerned with finding the shortest closed path joining n points where the distances between all pairs of points are given.

For a set of points in a metric space we establish that the shortest path will consist of a single loop circuit that will never cross itself. The problem is formally stated in Linear Programming terms. The difficulty in the Linear Programming formulation is to avoid the appearance of sub-cycles. This has been effected by making the programme dynamic. The computational cost of a complete enumeration of all possible circuits or the cost of solving the Linear Programme being prohibitive for a large number of points, we attempt a practicable computing routine. The method proceeds by writing down the set of distances as a square matrix and selecting from the matrix a set of n links which will form an initial single loop circuit: the circuit is changed by simultaneously removing three links and introducing three new links without destroying the single loop form of the circuit.

Related methods and problems are discussed.
Algebraic Theory of the Computing Routine for Tests of Significance on the Dimensionality of Normal Multivatiate Systems
Authors: Binet F. E. , Watson G. S.
Published: December 2018

Before attempting to allot individuals to one of a set of normal multivariate populations, it is usual to make preliminary tests of significance to discover the structural relationships of the populations. In order to secure this reduction there are two separate methods, due to Fisher and (essentially) to Wilks.

A simple algebraic proof is given for the identity of the statistics obtained from the two methods, and the condition under which Fisher’s method leads to less computation is pointed out.

To facilitate the contrast of the two methods, some well known principles and procedures have been restated in a more convenient form.
Partitions in More than One Dimension
Authors: Bennett J. H.
Published: December 2018

Several important properties are derived for the new partitional functions introduced by Fisher (1950) in his enumeration of the number of partitions of an integer in an arbitrary number of dimensions. These functions readily lead to the enumeration of the number of partitions of an integer in more than one dimension and with given marginal partitions. This enumeration is given for bipartitions of the integers 1 to 8 inclusive.
On the Estimation of Small Frequencies in Contingency Tables
Authors: Good I. J.
Published: December 2018

Suppose that we require estimates of the population frequencies corresponding to small entries in a large pure contingency table containing heterogeneity. It would be natural to lump rows or columns together with weights depending on the correlation coefficients. But if the rows, say, are nearly orthogonal, no lumping method will be reasonable. Moreover, if many of the entries in the table are missing, a lumping method may fail. A method is given here (which it may be possible to combine with a lumping method when appropriate) that depends on an assumption for the initial distribution of the “association factor” in each cell, i.e. the ratio of the population frequency in the cell to the product of the population frequencies of its row and column. If the logarithm of the association factor is assumed to have, initially, a normal distribution, then the final expectations and variances of the population frequencies and of their logarithms can be expressed in terms of the “after-effect” function which was originally tabulated by K. W. Wagner for an electrodynamical application. Instead of a log-normal distribution, a Pearson Type III distribution may be assumed for the association factor. This assumption may be less accurate but is easier to handle.

The paper concludes with a list of properties of the after-effect function (and of some of its generalizations) that may be useful for further tabulation.
The Within-Animal Bioassay with Quantal Responses
Authors: Claringbold P. J.
Published: December 2018

The four-point cross-over design is four times as economical in animals as the standard Allen-Doisy quantal bioassay.

This finding is applicable to other bioassays provided the total variation of response includes components due to animal differences and due to secular trends in response.

Analysis of quantal data with one animal per treatment group may, if the expected proportion responding is in the interval 5–95 per cent., be simplified by scoring the responses as 0 or 1, and performing an analysis of variance.
On Limiting Distributions Arising in Bulk Service Queues
Authors: Downton F.
Published: December 2018

The limiting behaviour is discussed of a simple queueing process, in which customers. arrive at random, form a single queue in order of arrival, and are served in batches, the size of each batch having a fixed maximum, as this maximum tends to infinity. Limiting distributions are obtained for the situations where the service interval has a χ2 distribution with an even number of degrees of freedom, and the queue is assumed to be in equilibrium. These situations include as special cases random service and regular service. A similarity is noted between these limiting results and the solution to other queueing situations.
The Efficiency of the Records Test for Trend in Normal Regression
Authors: Stuart Alan
Published: December 2018

A test for trend based on records is found to have asymptotic relative efficiency of zero compared with several other tests against normal regression alternatives, while two other well-known tests have zero efficiency compared even to the records test.
Some Experimental Designs of Use in Changing from One Set of Treatments to Another, Part I
Authors: Freeman G. H.
Published: December 2018

A description is given of various designs of use in changing from one set of treatments to another. Most of the designs are well known, but one, which involves new treatments partially balanced with respect to both blocks and original treatments, has not previously been described in general. The analysis of variance is given for this design, together with the variance of treatment differences, a lemma of use for the derivation of the latter also being proved. The analysis of covariance is given, this presenting no difficulties, and the design is also shown to be valid. An example is given, showing the derivation of all the parameters in an actual design.
Routine Analysis of Replicated Experiments on an Electronic Computer
Authors: Yates F. , Healy M. J. R. , Lipton S.
Published: December 2018

The paper describes the methods that have been developed for the analysis of replicated experiments on the electronic computer at Rothamsted. The general principle followed is to make the machine carry out the whole of the analysis, including the calculation of standard errors and residuals, the results being given in a form suitable for immediate transmission to the experimenter. Missing plots and covariance can also be handled. Preliminary computations on the raw data are carried out by the machine, using a routine which is common to all types of design. The data are punched on tape, a variate at a time, in the order they are received from the experimenter, i.e. without de-randomization.

Programmes have now been written for the commoner types of design such as randomized blocks, Latin squares, 33 (single replicate), and 2n designs. As an example, one of the more complex programmes, that for 2n designs, is described in some detail. This programme handles all types of 2n design containing up to 128 plots, including confounding, estimation of error from high order interactions, fractional replication, factors at 4 and 8 levels, and 8 × 8 quasi-Latin squares.

A brief outline of the mode of operation of electronic computers is given.
The Modified Latin Square
Authors: Rojas B. , White R. F.
Published: December 2018

The modified latin square design (semi-latin square) is described and discussed. The expectations of the mean squares are derived and, from these, the nature of the bias is developed. Empirical values of the bias are obtained from some corn uniformity data and found to be small. An unbiased estimator of the error of a comparison among treatments is obtained. A formula for obtaining the efficiency of the modified latin square relative to randomized blocks, using modified latin square data, is given and this formula is applied to the uniformity data. The efficiency of the modified latin square with this type of data is found to be of the order of lattice designs. The analysis of variance of a group of modified latin square designs performed at different “places” is discussed.
Statistical Approach to Problems of Cosmology
Authors: Neyman Jerzy , Scott Elizabeth L.
Published: December 2018

The basic hypothesis underlying current deterministic cosmologies, namely the so-called cosmological principle, can be stated precisely only in terms of probabilistic concepts. Consequently, considerable progress and aesthetic gain may be expected if determinism is abandoned and replaced by a frank probabilistic treatment of cosmology. This requires the adoption of the view that the Universe is a realization of a stochastic process which is stationary in the three (spatial) co-ordinates (cosmological principle) and possibly also stationary in the fourth (time) co-ordinate (“perfect” cosmological principle). Two examples are given of problems relevant to deterministic cosmological theories that elude deterministic treatment, but lend themselves to an indeterministic statistical study.
The Effect of the Size of the Waiting Room on a Simple Queue
Authors: Finch P. D.
Published: December 2018

The effect of the size of the waiting room on a queue with a random input process is considered, a particular case studied is that for which the service time is also random. The study originated in an investigation of storage problems along a production line and in an analysis of the provision of the internal buffer storage in electronic digital computers. For this reason the study is formulated in terms applicable to storage problems.
Formulae for Calculating the Operating Characteristic and the Average Sample Number of Some Sequential Tests
Authors: Kemp K. W.
Published: December 2018

Expressions and curves are given from which it is possible to calculate the operating characteristic and average sample number of a number of sequential sampling schemes to a close degree of approximation.
On Corrections to the Chi-Squared Distribution
Authors: Darwin J. H.
Published: December 2018

The method of characteristic functions is used to obtain a correction to the usual approximation to the distribution of the chi-squared goodness of fit criterion for a multinomial distribution.
The Behrens-Fisher Distribution and Weighted Means
Authors: James G. S.
Published: December 2018

(1) The Behrens-Fisher and Welch solutions of the two-means problem are compared, and the former shown to be lacking in sensitivity. (2) Yates’s and the author’s tests for a weighted mean are compared, and the former shown to reject the true value of the mean, in error, more often than is indicated by the nominal significance level (in extreme cases nearly twice as often). (3) A new test for weighted means is put forward which does not neglect the information on the variances supplied by the difference between the means, and which is “exact”. (4) An incidental by-product is the Cornish-Fisher expansion for a symmetrical distribution, taken to some further terms.
Some Simple Duration-Dependent Stochastic Processes
Authors: Mercer A.
Published: December 2018

The problem considered is that of two discrete states in continuous time; the transition probabilities depend on the total time, t, spent in, say, state 1. The probability that at a given time τ the process is in state i and the total time spent in state 1 lies in (t, t + δt) is derived. Some limiting cases for which the transition probabilities are powers of t are considered. The probability that the total time spent in state 1 reaches t in the interval (τ τ+ δτ) is also considered and some limiting distributions are obtained.
A Statistical Theory of Remnants
Authors: Aitchison J.
Published: December 2018

In certain manufacturing processes where varying lengths of material are produced and subsequently cut up into unit lengths the problem of loss from wasted material arises. In this paper we postulate a statistical model of such a process and study how and in what sense we may minimize this loss. Two simple tests of whether a process is under optimum control are proposed. Finally we consider briefly two extensions of the theory to corresponding problems in k dimensions.
Estimation of Location and Scale Parameters for the Rectangular Population from Censored Samples
Authors: Sarhan Ahmed E. , Greenberg B. G.
Published: December 2018

Best linear estimates of location and scale parameters under conditions of Type II censoring are obtained for the rectangular distribution. In addition, functions of the location and scale parameters such as the start and end of the distribution, as well as percentile points, are also estimable. A special case occurs where the rectangular distribution starts at the origin. Tables are available from the authors to furnish coefficients to facilitate all computations for samples of size up to 10. Graphs are included to show the effect on the relative efficiency caused by censoring.
The Output Process of the Queueing System M/G/1
Authors: Finch P. D.
Published: December 2018

It has been proved by Burke (1956) and Reich (1957) that the output process of a single server queue with a Poisson input process and negative exponential distribution for service time is again a Poisson process with the same parameter as the input process. In this paper single server queues with a Poisson input process and a general class of service time distribution are considered. It is shown that successive interdeparture intervals are independent in the limit only in the case the service time is negative exponential. It is shown also that if the size of the waiting room is taken into account as in Finch (1958) then successive departure intervals are not independent in the limit even when the service time is negative exponential.
Note on Fisher’s Transformation of the Correlation Coefficient
Authors: Laubscher N. F.
Published: December 2018

A family of modifications to Fisher’s transformation of the correlation coefficient is introduced. It is shown that Fisher’s form of the transformation is optimum within this family.
The Busy Period in Relation to the Single-Server Queueing System with General Independent Arrivals and Erlangian Service-Time
Authors: Conolly B. W.
Published: December 2018

The Busy Period relating to GI/Ek/1 is investigated by the method used by Conolly (1959) for GI/M/1. The system is simulated by a model in which arrivals take place in groups of fixed size k, service on each customer being random. This is an alternative to the more familiar “phase-service” concept used, for instance, by Wishart (1956).
A Note on Prediction from an Autoregressive Process Using Pistimetric Probability
Authors: Roy A. D.
Published: December 2018

The distributions of the parameters and of the future values of a Markov scheme are obtained by a technique which combines the use of Bayes' theorem with the method of inference sometimes considered appropriate to situations where there exist sufficient estimators.
Estimation of Parameters in Time-Series Regression Models
Authors: Durbin J.
Published: December 2018

We consider the estimation of the coefficients in a general linear regression model in which some of the explanatory variables are lagged values of the dependent variable. For discussing optimum properties the concept of best unbiased linear estimating equations is developed. It is shown that when the errors are normally distributed the method of least squares leads to optimum estimates. The properties of the least-squares estimates are shown to be the same asymptotically as those of the least-squares coefficients of ordinary regression models containing no lagged variables, whether or not the errors are normally distributed. Finally, a method of estimation is proposed for a different model which has no lagged dependent variables but in which the errors have an autoregressive structure. The method is shown to be efficient in large samples.
Maximum-Likelihood Estimation Procedures and Associated Tests of Significance
Authors: Aitchison J. , Silvey S. D.
Published: December 2018

The essence of many statistical problems, including most standard techniques, is to test whether or not the unknown parameters of an appropriate statistical model satisfy certain restrictions; and the outcome of such a test dictates whether it is necessary to provide estimates of these parameters which also satisfy the restrictions. In this paper we discuss and illustrate the relative merits, as practical tools for the consulting statistician, of two large-sample techniques of wide applicability to such situations: (i) unrestricted maximum-likelihood estimation with its associated Wald test, (ii) restricted maximum-likelihood estimation with its associated Lagrange-multiplier test. The discussion falls into two main sections corresponding to two methods of specifying restrictions, as constraint equations in the parameters, or as freedom equations expressing the parameters in terms of a second smaller set of parameters. The methods are modified by a simple device to apply to the case where constraints on the parameters are necessary to allow their identification.
Queueing at a Single Serving Point with Group Arrival
Authors: Conolly B. W.
Published: December 2018

Some aspects are investigated of a single-server queueing process in which customers arrive in groups, but are served individually. The input is of the “general-independent” type; service is negative exponential.
On Some Extensions of Bayesian Inference Proposed by Mr Lindley
Authors: Fisher R. A.
Published: December 2018

Critical comments are given on an extension of Bayesian inference proffered by Lindley (1957, 1958). The process is applied to the combination of two normal samples.
A Note on Tests of Homogeneity Applied after Sequential Sampling
Authors: Cox D. R.
Published: December 2018

It is pointed out that tests based on permutation theory can often be validly used to test the homogeneity of results obtained in a sequential trial.
Confidence Limits for Multivariate Ratios
Authors: Bennett B. M.
Published: December 2018

As a generalization of Fieller’s theorem for the case of two or more multivariate populations, confidence limits are derived for several ratio problems. These confidence limits are the shortest interval provided by the real roots of a certain high-order polynomial, the coefficients of which are subject to sampling variation. An example from anthropometry is given for the practical determination of these confidence limits.
The Time-Dependent Solution for an Infinite Dam with Discrete Additive Inputs
Authors: Yeo G. F.
Published: December 2018

This paper considers an infinite dam in discrete time fed by independent additive inputs and having a unit release at the end of each unit time interval, unless the dam is empty. The time-dependent solution for the dam content distribution is obtained. This depends on the probability of emptiness, which is itself determined as a function of the known probability of first emptiness. The general result is used to find solutions for the dam with negative binomial and Poisson inputs.
A Test of Homogeneity for Ordered Variances
Authors: Vincent Shirley E.
Published: December 2018

Suppose that an estimate of variance is available from each of a set of normal populations. The situation in which there is a definite trend in the magnitude of the population variances is considered, and this is represented by assuming the ith variance to be inversely proportional to λ + iμ. A uniformly most powerful unbiased test is developed for the hypothesis μ = 0, and methods for estimating μ and λ are considered.
A Simple Congestion System with Incomplete Service
Authors: Cox D. R.
Published: December 2018

Customers arrive at random in a system with m servers, each customer having a desired service-time. If on the arrival of a new customer all servers are busy, one of the m customers being served is displaced, even though his desired service-time is incomplete. The choice of a customer for displacement is specified by a displacement rule. Thus each customer has an achieved service-time, which may be less than the desired service-time. One index of efficiency of the system is the ratio I of the mean achieved service-time to the mean desired service-time. The quantity I is calculated for random arrivals under several different sets of assumptions concerning the displacement rule and the distribution of desired service-time. The problem was suggested by an investigation of systems of production control.
Moments of a Truncated Bivariate Normal Distribution
Authors: Rosenbaum S.
Published: December 2018

The moments are obtained for a bivariate normal distribution which is singly truncated with respect to both variables; the variables may be correlated. From these moments the parameters of the distribution can be estimated.
An Approach to the Scheduling of Jobs on Machines
Authors: Page E. S.
Published: December 2018

Two methods based on sorting procedures are suggested for scheduling jobs on machines. One is shown in series of simulated trials to yield substantially better results than an existing method with the same amount of computing, and the other to give about the same results with less computing.
On Quadratic Estimates of the Interclass Variance for Unbalanced Designs
Authors: Read Robert R.
Published: December 2018

The optimum properties of the popular estimate of the interclass variance in components of variance models are known if there are an equal number of observations in the subclasses. In contrast to this, it is shown that unbiased quadratic estimates having uniformly minimum variance do not exist if the design is not balanced.
Some Examples of Bayes' Method of the Experimental Determination of Probabilities a Priori
Authors: Fisher Ronald
Published: December 2018

Some further examples are given of Bayes' method of determining probabilities a priori by an experiment.
On the Asymptotic Sufficiency of Certain Order Statistics
Authors: Rohatgi Mandakini S.
Published: December 2018

The problem of estimating distance to a stellar system, from measurements on the apparent magnitude of the jth brightest object in the system, reduces to estimating the location parameter of a distribution. Astronomical observations provide apparent magnitudes of only a few of the brightest objects in a system. In this connection, two theoretical problems arise. One is to find the best estimate based on available observations, and the other to determine which of the objects should be observed in order to attain the greatest precision for the estimate. It is found that, if the number of objects n in the system is large, then, depending on the luminosity function, either the apparent magnitude of the brightest or the faintest of the available objects is asymptotically sufficient for the location parameter.
The Estimation of Slope When the Errors are Autocorrelated
Authors: Jenkins G. M. , Chanmugam J.
Published: December 2018

In this paper an investigation has been made of the problem of constructing systematic and randomized designs for the estimation of slope when the errors are autocorrelated. Some results are obtained for general stationary processes and specific designs given for moving average and autoregressive processes. Comparison is made with randomized block designs. The theory is illustrated on a time-series obtained from a uniformity trial on a batch production process.
Large-Sample Restricted Parametric Tests
Authors: Aitchison John
Published: December 2018

Restricted tests are designed to examine a hypothesis not against the natural complete alternative but against some subhypothesis of that alternative. The theory of such tests provides a unification of many aspects of large-sample tests—the analysis of nested hypotheses, the possibility of using tests of increased power and of easing certain computational procedures, the notions of separable hypotheses and of partitioning a test statistic. While many standard results can thus be brought within the framework of this theory the emphasis throughout the paper is on providing the consulting statistician with manageable tools for non-standard situations.
On a Simple Procedure of Unequal Probability Sampling Without Replacement
Authors: Rao J. N. K. , Hartley H. O. , Cochran W. G.
Published: December 2018

A simple procedure of unequal probability sampling without replacement is proposed. It leads to an estimator of the population total having a smaller variance than is obtained by sampling with replacement. Other advantages of the present method are simplicity of calculation and the possibility of estimating exactly the variance of the estimator.
Sampling Properties of Tests of Goodness-Of-Fit for Linear Autoregressive Schemes
Authors: Chanda K. C.
Published: December 2018

Sampling properties of tests for trend-free linear autoregressive schemes due to Bartlett and Diananda, to Quenouille, and to Whittle have been investigated. The investigation consists of the calculation of the first two moments of the various test-criteria which are χ2‘s with k degrees of freedom, where k is the number of asymptotically independent linear compounds on which the χ2‘s are based. The purpose is to examine the effect of the combination of individual constituents of χ2 on the validity of the test-criteria when the sample is not large. It has been demonstrated that when the sample is not very large, the contributions due to the terms involving k/n are considerable, a fact which confirms the empirical conclusions of Rao and Som (1951). In view of the apparent loss of validity thus entailed a modified test is suggested to be used when the sample size is small.
Analysis of Stationary Processes with Mixed Spectra—Ii
Authors: Priestley M. B.
Published: December 2018

We investigate the asymptotic power functions of three types of tests for detecting the presence of harmonic components superimposed on a nonuniform spectral density function. These power functions are tabulated for a range of values of the signal to noise ratio of the process, and each test is applied to several artificial series of the “mixed ”type with known structure.
The Use of a Generalized Multinomial Distribution in the Estimation of Correlation in Discrete Data
Authors: Tallis G. M.
Published: December 2018

This paper presents a joint distribution for n identically distributed multinomial variates, Xq. The distribution is constructed so that C(Xq, Xr) = pV(X) for all q and r, q≠r, and applications of this model to the estimation of correlation in discrete data are discussed.
Tests of Goodness of Fit
Authors: Anscombe F. J.
Published: December 2018

An explicit account is given of a procedure for assessing goodness of fit of some observations with a hypothesis, generally known as a “test of significance”; the description is close in spirit to R. A. Fisher’s original conception. The relation of this test procedure with Bayesian procedures and with the Neyman–Pearson theory of tests is discussed. Some very tentative suggestions are made regarding the choice of a test criterion and the use to which the result of the test can be put. The role of conditional sampling distributions is studied. Various examples are discussed. In particular, a brief exploration is made in the apparently virgin territory of testing goodness of fit when the observations have been taken according to a sequential rule.
The Distribution of the Time-To-Emptiness of a Discrete Dam under Steady Demand
Authors: Mott J. L.
Published: December 2018

The distribution of the time-to-emptiness of a dam with integral random inputs and steady unit demand is derived from a simple combinatorial lemma.
On the Normal Stationary Process—Areas Outside Given Levels
Authors: Leadbetter M. R.
Published: December 2018

Let x(t) be a normal stationary process, observed for 0 ≤ t ≤ T, and c a given positive constant. Formulae are derived for the mean and variance of the area above the level + c formed by the process x(t) in (0, T). Corresponding results are obtained for the sum and the difference of the areas above the level + c and below the level — c.
Maximum Likelihood in Three-Way Contingency Tables
Authors: Birch M. W.
Published: December 2018

Interactions in three-way and many-way contingency tables are defined as certain linear combinations of the logarithms of the expected frequencies. Maximum-likelihood estimation is discussed for many-way tables and the solutions given for three-way tables in the cases of greatest interest.
The Spectral Analysis of Point Processes
Authors: Bartlett M. S.
Published: December 2018

The spectral analysis of stationary point processes in one dimension is developed in some detail as a statistical method of analysis. The asymptotic sampling theory previously established by the author for a class of doubly stochastic Poisson processes is shown to apply also for a class of clustering processes, the spectra of which are contrasted with those of renewal processes. The analysis is given for two illustrative examples, one an artificial Poisson process, the other of some traffic data. In addition to testing the fit of a clustering model to the latter example, the analysis of these two examples is used where possible to check the validity of the sampling theory.
A Correction to “The Solution of Queueing and Inventory Models by Semi-Markov Processes”
Authors: Fabens A. J. , Perera A. G. A. D.
Published: December 2018

An incorrect assumption of independence produced an error in the waiting time distributions derived in Fabens (1961). The correct distributions are derived here.
Local Conditional Sufficiency
Authors: Fraser D. A. S.
Published: December 2018

The concept of sufficiency can be examined locally for a neighbourhood of a parameter point θ0 and it leads to an estimate that has minimum variance among locally-unbiased estimates (Fraser, 1964). This local estimate generates the natural global estimate for the case of the simple exponential model, but it does not do so for the important case of the general translation model. The global estimate, the Pitman estimate, for this translation model is rooted in ancillarity and conditional sufficiency. In this paper the concept of local conditional sufficiency is developed; the asymptotic distribution of the conditionally-sufficient statistic is investigated for large samples; and for the maximum-likelihood estimate a conditional large-sample variance is obtained which differs from that usually used.
Bayesian Estimation of the Variance of a Normal Distribution
Authors: Evans I. G.
Published: December 2018

Some new general forms of estimators of the variance of a normal distribution are derived using Bayesian methods, and the conditions under which they lead to previously proposed estimators are discussed.
Some Results for Fixed-Time Traffic Signals
Authors: Buckley D. J. , Wheeler R. C.
Published: December 2018

An investigation is made of the queues and delays which occur when a Poisson traffic stream is subjected to a blockage. It has been assumed that the traffic is free-flowing when the blockage commences, that when the blockage is terminated delayed vehicles depart with constant time separations, and that no queueing occurs after the system has emptied for the first time. A blockage of duration ρ causes j vehicles to be delayed a total time w. Expressions are obtained for the Laplace transform of the joint distribution of j and w, the marginal distribution of j and the Laplace transform of the marginal distribution of w. The expected delay at a fixed-time traffic signal is then obtained for the case where there is a negligible probability that vehicles are still queueing at the end of the green phase.
On Local Inference and Information
Authors: Fraser D. A. S.
Published: December 2018

Two techniques for local unbiased estimation have been developed in recent papers (Fraser, 1964a, b). In this paper these techniques are used to examine some aspects of inference and information. Both techniques show inadequacies for translation-invariant models, and this despite the development of the second technique on the basis of conditional sufficiency which is basic to translation-invariant models. Inference and information are then examined briefly for translation-invariant models and some modified definitions of information are proposed. The results are then extended to other models and lead to definitions of local structural probabilities on the parameter space and to a residual likelihood function.
On the Power of Two-Sample Rank Tests on the Equality of Two Distribution Functions
Authors: Gibbons Jean D.
Published: December 2018

This paper is a comparative study of the power functions of two-sample rank tests of the hypothesis of equal distributions, H  0: H = G. Numerical results are given for the alternative of two extreme distributions, H1:H=1−(1−F)k,    G=Fk, with F an unspecified continuous cumulative distribution function, k = 2, 3, 4, sample sizes m = n ≤ 6, significance levels 0·01, 0·05, 0·10. Randomized decision rules are employed when necessary to attain the exact significance level. Some results are also given for unequal sample sizes. The tests considered are the most powerful rank test, Terry's test, one- and two-sided Wilcoxon tests, one- and two-sided median tests, the Wald-Wolfowitz runs test, and two new tests, called the Gamma and Psi tests. These are derived herein as the locally most powerful rank tests against slight variations of the alternative of two extreme distributions. Comparisons are made with the power of these same tests against normal alternatives differing only in location.
The Detection of Partial Association, I: The 2 × 2 Case
Authors: Birch M. W.
Published: December 2018

A criterion is derived for testing null hypotheses of conditional independence of two dichotomous random variables. This test has optimum properties when the alternative hypothesis is that the association of the two random variables in the conditional distribution is, in a certain sense, constant.
Spectral Analysis in the Presence of Variance Fluctuations
Authors: Herbst Laurence J.
Published: December 2018

The present paper is concerned with the following questions: (1) If a random sequence Xt (t = 0, ± 1, ± 2, …) is not second-order stationary, can one meaningfully define a spectral density function ? (2) If so, can one hope to consistently estimate it ? For the special case of a random sequence of form Xt=∑0pajσt−jηt−j, where the ηt are independently, normally, distributed with zero means and unit variances and the aj and σt-j are non-random finite quantities, it is argued that the answers to both questions are affirmative provided σt2 (t = 0, α 1, α 2, …), while not, in general, constant, do satisfy the requirements σt2 ≤ Q < ∞ (t = 0, α 1, + 2, …), where Q does not depend on t, and N−1∑1Nσt2→γ0<∞ as N → ∞.
Note on a χ2-Approximation for the Multivariate Sign Test
Authors: Bennett B. M.
Published: December 2018

Certain properties of the multivariate normal integral were employed by Bennett (1962) to propose a sign test for the equality of means in samples of n observations from two correlated p-variate populations for the cases p = 2, 3. Using the multinomial distributions for n = 10, 20, the exact distribution and significance points have been computed for the case p = 2, and compared with that of an approximate χ2-test and also that of the suggested modification by Wise (1963) of the χ2-test, when now maximum-likelihood estimates of the multinomial probabilities are used.
A Further Result on the Relevance of the Dispersion of a Radon–Nikodym Derivative to the Problem of Measuring Association
Authors: Ali S. M. , Silvey S. D.
Published: December 2018

Let x and y be jointly distributed random variables (not necessarily real-valued) and let φ be the Radon–Nikodym derivative of their joint distribution with respect to the product of their marginal distributions. If x′ and y′ are functions of x and y respectively and φ′ is a corresponding derivative for x′ and y′, then the dispersion of φ, as determined by the product of the marginal distributions of x and y, is not less than that of φ′, this being true in the strongest possible sense.
Spacings
Authors: Pyke R.
Published: December 2018

The distribution theory and possible constructions for spacings (differences) between successive order statistics of a sample of independent and identically distributed random variables are reviewed. The problem of finding the limiting distribution of a function of spacings is discussed and some general limit theorems for such are presented.
The Numerical Solution of Some Non-Linear Equations, Useful in the Design of Experiments
Authors: Albasiny E. L.
Published: December 2018

This paper investigates the numerical solution of non-linear equations arising in experimental design problems of Draper and Lawrence (1965).
Additional Results Concerning Estimable Functions and Generalized Inverse Matrices
Authors: Searle S. R.
Published: December 2018

A method is derived for calculating the unbiased estimator of any estimable function of the parameters of a linear model. its variance and its covariance with the estimator of any other estimable function are also obtained. An expression is also given for the F-value required in testing a general linear hypothesis.
Certain Properties of Gaussian Processes and Their First-Passage Times
Authors: Mehr C. B. , McFadden J. A.
Published: December 2018

The class of Gauss-Markov processes is characterized by their covariances. A functional equation is solved, giving the class of all Gauss–Markov processes with stationary transition probabilities. The notion of a conditionally Markov Gaussian process is introduced. A functional equation is solved which characterizes the class of stationary, conditionally Markov Gaussian processes. A class of non-stationary conditionally Markov Gaussian processes is constructed. Using a transformation of the Wiener process, an explicit expression is obtained for the first-passage-time density function, from level x  0 ≠ 0 at time t  0 to the level zero at time t > t  0, for all Gauss–Markov (G.M.) processes. For two particular classes of G.M. processes an explicit expression is obtained for the density of the first-passage time, from any level x  0 at t  0 to any level x ≠ x  0 at time t > t  0. Some of the results are extended to a class of Gaussian processes which are not Markovian.
Allocation Rules and Their Error Rates
Authors: Hills M.
Published: December 2018

The problem of choosing allocation rules and assessing their errors of mis-allocation is formulated in general terms. An inequality connecting the average apparent, the optimum, and the average actual error is established. The distributions of the actual and apparent errors are considered in some special cases. The estimation of the average actual error is also considered.
On Efficient Multinomial Estimation
Authors: Bhat B. R. , Kulkarni N. V.
Published: December 2018

The problem of obtaining efficient estimators and sampling plans for a parametric function of a multinomial parameter is studied, and conclusions similar to those of Degroot (1959) and Girshick et al. (1946) for the binomial case are reached.
An Experimental Study of Certain Screening Processes
Authors: Finney D. J.
Published: December 2018

The paper discusses alternative procedures for successive selections from a population aimed at maximizing the mean value of the members finally selected. At each stage, errors of measurement obscure the attempt to select optimally, the magnitude of the errors being dependent both upon the proportion of the original population still surviving and the allocation of a part of the total available resources to measurement at the particular stage. Selection of a finite number of members at each stage is compared with selection of fractions from an infinite population, and the general trends are found to be similar. For all but the most trivial specifications of the selection, mathematical analysis is hopelessly intractable. A great amount of numerical study has been undertaken and is summarized in extensive tables. Some broad conclusions emerge, notably that little can be gained by going beyond three or four stages (with fixed total resources), and that both the selection fraction and the proportion of resources employed should be kept approximately constant from stage to stage.
An Extension of the Triangular Association Scheme to Three Associate Classes
Authors: John Peter W. M.
Published: December 2018

The treatments in a triangular association scheme may be represented as ordered pairs of integers. This representation is extended to ordered triplets to give a scheme with three associate classes. The latent roots of the matrix NN', where N is the incidence matrix, are calculated and are used to establish the non-existence of some designs. Several series of designs are given together with their efficiencies.
On the Evaluation of Probabilities of Convex Polyhedra Under Multivariate Normal and t Distributions
Authors: John S.
Published: December 2018

It is noted that probabilities of convex polyhedra in p-dimensional space can be expressed in terms of a function V(h1, h2, …, hp), if the distribution is normal, and in terms of a function Vn(h1, h2, …, hp), if the distribution is the t-distribution with n degrees of freedom. Formulae expressing V(h1, h2, …, hp) and Vn(h1, h2, …, hp) in terms of similar functions of lower order are obtained. An asymptotic expansion of Vn(h1,…, hp) in powers of n-1 is also included. The final section contains formulae connecting Vn(h1, h2, …, hp) with Vn±2(h1, K2, …, hp).
Some New Results on the Distribution of the Sample Correlation Coefficient
Authors: Ruben Harold
Published: December 2018

A simple approximate normalization for the correlation coefficient in normal samples is obtained which is shown to compare favourably with previous normalization approximations (the usual large-sample normalization, Fisher's hyperbolic tangent transformation and Hotelling's improvement of that transformation). A second approximation, based on F and valid for large n | ρ | (n = sample size, ρ = population correlation), is also obtained. The results follow from a representation of the correlation in normal samples as the ratio of a linear combination of a standardized normal deviate and a Xn-1 to a Xn-2, where the three latter variates are independent.
On the Independence of Quadratic Forms
Authors: Shanbhag D. N.
Published: December 2018

A correction is made to a paper by Good (1963) on the independence of quadratic forms.
The Use of the Concept of a Future Observation in Goodness-Of-Fit Problems
Authors: Guttman Irwin
Published: December 2018

An attack on the problem of goodness of fit is made by combining a Bayesian and sampling argument; the Bayesian part is effected by using the distribution of a future observation, while the sampling argument concerns itself with the distribution of a “chi-squared like” statistic, which measures discrepancies of observed frequencies from those predicted by the distribution of the future observation. Examples are given for the case of sampling from the binomial, Poisson and normal distributions. An interesting application arising from the above approach is a procedure for estimating the degree of a polynomial response function.
On Fixed-Width Confidence Bounds for Regression Parameters and Mean Vector
Authors: Srivastava M. S.
Published: December 2018

In this paper Chow and Robbins's (1965) sequential theory has been extended to construct a confidence region with prescribed maximum width and prescribed coverage probability for (i) the linear regression parameters, and (ii) the mean vector of a multivariate population. Except for the finiteness of the unknown variance and covariances in the respective two cases, no assumption regarding the form of their distribution functions is made. This result was announced in Srivastava (1966a). For some related results, refer to Gleser (1965), Srivastava (1966b) and Starr (1966).
Formal Expressions Which Can be Used for the Determination of the Operating Characteristic and Average Sample Number of a Simple Sequential Test
Authors: Kemp Kenneth W.
Published: December 2018

In this paper expressions are developed which are useful for determining the characteristics of a sequential test designed to test a simple hypothesis against a simple alternative. Simple equations are derived which give upper and lower limits for the operating characteristic and average sample number of the test in situations where formal expressions for these are not available.
On a Special Subset Giving an Irregular Fractional Replicate of a 2N Factorial Experiment
Authors: Banerjee K. S. , Federer W. T.
Published: December 2018

A special class of fractional replicates of 2n factorial experiments has been studied in this paper. Such fractional replicates are expected to be useful in practice where, for very large values of n, we may be interested only in the main effects. Analysis of such designs is simple in that the entire computational procedure is reducible to a simple routine.
A Bayesian Significance Test for Multinomial Distributions
Authors: Good I. J.
Published: December 2018

A Bayesian significance test for multinomial distributions is discussed, together with the results of 18 numerical experiments in which the test is compared with non-Bayesian methods. Several different non-Bayesian criteria are considered because the circumstances under which their tail-area probabilities can be conveniently approximated differ from one to the other. A provisional empirical formula, connecting the Bayes factors with the tail-area probabilities, is found to be correct within a factor of 6 in the 18 experiments. As a by-product, a new non-Bayesian statistic is suggested by the theory, and its asymptotic distribution obtained. It seems to be useful sometimes when chi-squared is not, although chi-squared also has a Bayesian justification for large samples.

The work originated in, and is relevant to, the problem of the estimation of multinomial probabilities, but significance tests are a better proving ground for assumptions concerning the initial (prior) distribution of the physical probabilities.
On a Class of Gaussian Processes for Which the Mean Rate of Crossings is Infinite
Authors: McFadden J. A.
Published: December 2018

A stationary Gaussian process X(t) is defined as locally Markov if the slope of its autocorrelation function has a finite jump at the origin. For such a process the mean rate of crossings at an arbitrary level a is infinite. For small (t2 − t1), the probability that X(t1) − a and X(t2) − a have opposite signs is calculated, both with and without the auxiliary condition, X(t3)∈ (x3, x3+dx3], …, X(tn+2)∈(xn+2, xn+2+dxn+2], where the times t3,…, tn+2 are all outside [t1,t2]. By inversion, it is shown that the conditional probability of this multiple event in X(t3), …, X(tn+2), given the sign change in (t1,t2), is asymptotically equal to the conditional probability of the same multiple event, given that X(t2) ϵ (a, a + da]. Again for small (t2 − t1). the probability that X(t) - a has one or more zeros is calculated, both marginally and conditionally. This event includes the case of a positive even number of crossings, which is not negligible. The above asymptotic equivalence between conditional probabilities can be extended also to this condition of one or more crossings. As an application, the probability of crossings in two small intervals (at different levels) is considered.