bluuebunny/crossref_metadata_2025 huggingface dataset:
The Detection of Linkage in Human Genetics
Authors: Smith Cedric A. B.
Published: December 2018

There are three methods used in practice to detect human linkage. These are: (i)The sib-pair method of Penrose, applicable to pairs of brothers and sisters, when the parents’ genotypes are unknown or disregarded.(ii)The method of efficient scores (u-functions), due to Fisher and Finney, applicable to families consisting of 2 or more children, together with either or both genotypes of the parents if known.(iii)The method of likelihood ratio, or backward odds, due to Haldane & Smith, and most suitable for large pedigrees: this links up with maximum likelihood estimation.

Each method has its own advantages and disadvantages. It is part of the purpose of this paper to present a connected account of the subject, showing the relations between the three methods. It appears that all are really different forms of the likelihood ratio test, suitably adapted for practical use.

It is suggested that recessive children of cousin parents can be used to detect linkage.

Methods of using sib-pair tables to calculate efficient scores are given. As the sib-pair table is convenient in use, methods are given to extend it to families of more than 2 sibs, though this may not be fully efficient. It is also pointed out that tables of lods (log odds) can be prepared for use with families of 2 or 3 sibs, and an example is provided.
The Estimation of the Parameters of a Birth and Death Process
Authors: Moran P. A. P.
Published: December 2018

In a previous paper the estimation of the sum λ + μ of the parameters of a simple birth and death process was considered. In the present paper we consider the estimation of λ(λ + μ)−1. This problem is equivalent to the problem of estimating the probabilities of steps to the right and left from an observed realization of a random walk which has one absorbing boundary and which is terminated, if necessary, after a preassigned number of steps. The properties of various estimators are considered.
Selection without Replacement from within Strata with Probability Proportional to Size
Authors: Yates F. , Grundy P. M.
Published: December 2018

In selection with probability proportional to size x from within strata without replacement, the usual method of selection gives rise to bias in the estimate of the total of a variate y derived by weighting the units by weights proportional to 1/x. By means of numerical examples it is shown that the amount of this bias is usually quite trivial. If, however, unbiased estimates are required, the true total probabilities of selection of the different units can be calculated easily for samples of 2, and with considerably more labour for samples of 3.

The bias in the ordinary formula for the estimation of error is also investigated, and the formula is shown to be reasonably accurate. Horvitz and Thompson have given an unbiased estimator of the error variance, but this is shown to be inefficient and a new unbiased estimator is given.

A method of revising the size measures so that with the usual method of selection the true total probabilities of selection are proportional to the original size measures is given for samples of 2. Horvitz and Thompson’s solution of this problem does not appear to give satisfactory approximations in the cases met with in practice.

The selection of successive members of a sample with arbitrary sets of probabilities chosen solely so that the total probabilities shall be proportional to the original size measures, which has been advocated in various quarters, is criticized.
Some Results in Sampling Theory when the Units are Selected with Unequal Probabilities
Authors: Durbin J.
Published: December 2018

The main purpose of the present paper is to give a rule for calculating estimates of sampling error which can be applied to multi-stage designs of any degree of complexity. The rule is an extension of that given by Yates (1949) for the case of equal probabilities of selection. The paper goes on to discuss the relation between the theories of sampling with and without replacement and two approximate procedures are described which are easy to apply but lead to slight overestimates of sampling error.
Poor Man’s Monte Carlo
Authors: Hammersley J. M. , Morton K. W.
Published: December 2018

Monte Carlo methods often are and always should be part of the normal stock-in-trade of the ordinary practising statistician. This paper is an introduction to such methods, with an emphasis on those that require little or no calculating machinery. Three worked numerical examples, arranged in order of increasing difficulty, illustrate some of the more important general precepts. The last of these examples touches on some problems of self-avoiding walks, i.e., stochastic processes without multiple points.
The Application of Automatic Computers to Sampling Experiments
Authors: Tocher K. D.
Published: December 2018

This paper describes some of the problems that arise when automatic computers are used for conducting sampling experiments. The generation of random elements is discussed in detail and some methods of producing random variables with common distributions are described. The use of sampling methods to evaluate multivariate integrals is discussed. Finally a programme is devised to conduct a special form of a restricted random walk.
Transposed Branching Processes
Authors: Hammersley J. M. , Morton K. W.
Published: December 2018

We shall show in this paper how, in studying the growth-rate of a deterministic branching process, a simple change of viewpoint, which is quite trivial theoretically, may nevertheless lead to surprising computational simplifications; and, in the course of this, we shall indicate a basis on which the generated population can sometimes be reclassified into fewer classes.
A Further Approximation to the Distribution of Wilcoxon’s Statistic in the General Case
Authors: Sundrum R. M.
Published: December 2018

In an earlier paper, the author obtained the variance of Wilcoxon’s two-sample test statistic and used it to find the large sample power of Wilcoxon’s test against certain alternatives. In this paper general expressions are derived for the third and fourth moments of the statistic; these are found to depend on relatively few parameters. A short table is given of the values of these parameters under normal alternatives.
A Continuous Time Treatment of a Simple Queue using Generating Functions
Authors: Bailey Norman T. J.
Published: December 2018

In discussing the behaviour of a simple queueing process for continuous time, it is shown that results, previously obtained in a rigorous way by Ledermann and Reuter only by using a complicated spectral analysis, may in fact be derived in a comparatively simple way by means of a standard generating function technique.
An Outline of Linear Programming
Authors: Vajda S.
Published: December 2018

Linear Programming is defined and examples of its applications are given. Methods are indicated for the solution.
Statistical Concepts in Their Relation to Reality
Authors: Pearson E. S.
Published: December 2018

This paper contains a reply to some criticisms made by Sir Ronald Fisher in his recent article on “Statistical Methods and Scientific Induction”.
A Note on the Periodogram of the Beveridge Wheat Price Index
Authors: Gower J. C.
Published: December 2018

The complete periodogram of the Beveridge Wheat Price Index is given and is then used to test the goodness of fit of the second order autoregressive scheme proposed by Sargan.
Random Queueing Processes with Phase-Type Service
Authors: Jackson R. R. P.
Published: December 2018

A queueing process is studied in which customers arrive at random and are served in order of arrival at each of a number of counters arranged in series. Each counter of the series is permitted to have “many servers”, and the distribution of service time for a customer at each phase of service (e.g. at each counter) is supposed to be negative exponential. The equilibrium solution for the general case is obtained, but more detail is given for the single server system.
Fiducial Distributions and Prior Distributions: An Example in Which the Former Cannot be Associated with the Latter
Authors: Grundy P. M.
Published: December 2018

A class of one-parameter distributions is specified, for which the sample total is a sufficient statistic in samples of arbitrary size. It is proved that the resulting fiducial distribution of the parameter does not coincide with the distribution, a posteriori, given by Bayes' theorem, for any prior distribution whatever.
A Comparison of Two Sorts of Test for a Change of Location Applicable to Truncated Data
Authors: Barton D. E.
Published: December 2018

The powers of one and two-sample tests of change of centre of location based on the difference of sample quantiles are compared with those of the corresponding combinatorial tests based on quantile division of the sample. The powers are found to be asymptotically equivalent. Computation of particular cases suggests that for quite small samples there is good agreement of the powers of the two kinds of test with each other and with their common limiting form.
Analysis of Variance as an Alternative to Factor Analysis
Authors: Creasy Monica A.
Published: December 2018

An analogy between factor analysis and analysis of variance is pointed out, and an approximate test of significance is indicated. It is suggested that considerable simplification in the factor loadings may be made without appreciably affecting the final results. The interactions of factors is discussed, and it is suggested that batteries of tests should be designed so that they can be measured and tested. Some examples are given to illustrate the points made.
Stationary Distributions of the Negative Exponential Type for the Infinite Dam
Authors: Gani J. , Prabhu N. U.
Published: December 2018

The paper begins with a formulation of the infinite dam equations for the stationary distribution of the sum of the dam content and input, in both the discrete and continuous cases. Methods of solving these are briefly outlined, that for the discrete dam being due to Bailey (1954), while a second for the continuous dam is due to Lindley (1952).

In the continuous case, an input distribution of the negative exponential type is seen to result in a negative exponential stationary distribution of the dam content, together with a discrete probability that the dam be empty. It is shown that several other input distributions which are of approximately Gamma type below a specified value of the random variable, and with negative exponential tails, also give rise to similar negative exponential distributions of the dam content, with different discrete probabilities that the dam be empty.

The paper ends with a numerical comparison of the stationary probabilities for the content of finite and infinite dams in two simple cases of these input distributions.
The Matching Distributions: Poisson Limiting Forms and Derived Methods of Approximation
Authors: Barton D. E.
Published: December 2018

(1) The Matching Problem is reviewed. (2) Stevens’s approach is generalized to yield the explicit results so far known. (3) A continuous range of Poisson limits is derived for the various matching distributions by an elementary method (alternative to Kaplansky’s symbolic proof). (4) The approximations afforded by some of these are improved by orthogonal polynomial fitting. (5) Further Poisson limits are derived for certain general classes of multiple matching distributions; the asymptotic form for the number of Latin Rectangles is derived as a corollary and some numerical comparisons made. (6) Levene’s matching problem is generalized: A general Poisson limit is obtained and some exact distributions derived. Kullback’s matching distribution is generalized and a general Poisson limit obtained.
A Difference Equation Technique Applied to the Simple Queue with Arbitrary Arrival Interval Distribution
Authors: Conolly B. W.
Published: December 2018

In a previous paper (Conolly, 1958) a difference equation technique was applied to the simple, single-server queue with negative exponentially distributed service and arrival intervals. The technique is here generalized and illustrated in the case where arrival intervals are arbitrarily distributed.
A Simplified Model for Delays in Overtaking on a Two-Lane Road
Authors: Tanner J. C.
Published: December 2018

A theoretical formula is found for the average speed achieved over a long journey by a vehicle travelling along a road according to certain rules governing overtaking, when the other traffic on the road is postulated to behave in a particularly simple way.
Geometric Distributions in the Theory of Queues
Authors: Winsten C. B.
Published: December 2018

First a new explanation in probability terms is given for the occurrence of geometric or conditional geometric distributions in some queueing problems. The methods used allow some simple but useful generalizations to be obtained. In §§2 and 3, these results are used to find the equilibrium distribution for a queueing system for which customers can arrive late, and for which the service time has a negative exponential distribution. The effects of different assumptions about the distribution of lateness on the queue distribution are investigated. In §4, a generalization of a result in the theory of Markov chains given in §1 is derived.
Cyclic Queues with Feedback
Authors: Finch P. D.
Published: December 2018

Simple queues, with customers arriving at random for service at a number of service stations in series, are considered. Feedback is permitted either by feedback from the terminal server, or by feedback from each server of the series to the queue waiting for service at that stage. A constraint is introduced by imposing an upper limit on the number of customers in the system at any time. The service time of each server is supposed to be negative exponential.
Maximizing a Function in a Convex Region
Authors: Zoutendijk G.
Published: December 2018

The problem is considered of maximizing a function in a convex region. to solve this problem a new method is developed, to be called “method of feasible directions”. It is a method of steep ascent. Starting with a feasible solution a sequence of feasible trial solutions with ever-increasing values for the objective function is obtained. This sequence will converge to a (local) maximum solution in all cases of importance. to obtain a new trial solution from an old one we have to determine (a) a direction in which the objective function increases and the solution remains feasible and (b) the length of the step to be taken in this direction. The method of feasible directions is first developed for problems with linear constraints and applied to the special cases of a linear and a quadratic objective function. In the latter two cases the number of steps is finite. An extension of the method to problems involving non-linear constraints is also considered.
The Distribution of Vacancies on a Line
Authors: Page E. S.
Published: December 2018

From a line of n points, pairs of adjacent points are selected at random until only isolated single points remain. The mean and other moments of the distribution of the number of points remaining are found and some related problems are studied.
The Comparison of Regression Variables
Authors: Williams E. J.
Published: December 2018

Hotelling's test of significance for difference in efficiency of predictors is reformulated in terms of regression analysis. A test proposed by Healy is shown to differ from Hotelling’s test in general.
The Most Economical Sequential Sampling Scheme for Inspection by Variables
Authors: Wetherill G. B.
Published: December 2018

The two-point prior distribution of Barnard (1954) is used and leads to a method for calculating most economical sampling schemes for variables. An example is worked out where observations on items are normally distributed, and an application is given to determining most economical sampling schemes for group sequential sampling for attributes.
The Analysis of Exponentially Distributed Life-Times with Two Types of Failure
Authors: Cox D. R.
Published: December 2018

A number of alternative probability models are considered for the interpretation of failure data when there are two or more types of failure. Some of the statistical techniques that can be used for such data are illustrated on an example discussed recently by Mendenhall and Hader.
Regression Analysis When There is Prior Information About Supplementary Variables
Authors: Cox D. R.
Published: December 2018

In estimating the regression coefficient of y on x, we can sometimes increase precision by using supplementary observations, u. We need to have prior information about the form of the relations among y, x, and u. An implication for experimental design is noted.
On the Distribution of the Weighted Difference of Two Independent Student Variables
Authors: Ruben Harold
Published: December 2018

The distribution of the d-statistic, the weighted difference of two independent t-variables, is first obtained in integral form. This allows d to be interpreted as the ratio of two independent random variables, the numerator of which is a Student variable and the denominator a function of a Beta variable. It is suggested that the simplicity of the above integral form should enable extensive computation and tabulation of the d-statistic to be effected. Explicit forms for the distribution of d as finite or infinite series involving Incomplete Beta Function Ratios and Incomplete Gamma Function Ratios are derived for the special cases when (i) the degrees of freedom and weights of the component variables are equal and (ii) one of the components is normal.
A Problem of Delayed Service—I
Authors: Ben-Israel A. , Naor P.
Published: December 2018

A system is considered possessing the following characteristics: A repairman tends a number of machines—liable to break down from time to time—and some of his time is possibly taken up by ancillary duties, inspection, walking, etc. It is assumed throughout this study that the time consumed by attendance to a particular machine is independent of the state of the machine—working or idle. On the basis of this assumption a variety of models is discussed differing among themselves by the following considerations: (a)successional as against simultaneous attendance to machines;(b)constant Poisson breakdown intensity of the whole system as against constant Poisson breakdown intensity of individual machines in working order;(c)different assumptions regarding attendance time distributions. Expressions are developed for the mean, variance and, in some cases, the functional form of the distribution of  (i)the number of idle machines as observed by a random controller;(ii)the number of idle machines as observed by the repairman during one round; and(iii)the waiting time of an idle machine.
A Problem of Delayed Service—Ii
Authors: Ben-Israel A. , Naor P.
Published: December 2018

The system analysed in the preceding paper under the two alternative extreme assumptions of successional as against simultaneous attendance to machines is reconsidered under a “mixed” assumption: Machines are attended to in groups; repair of machines belonging to the same group is simultaneous, but different groups are dealt with in succession. Expressions are developed for the mean and variance of the number of idle machines as observed by a random controller and as observed by the repairman during one round.
A Method of Estimation of Missing Values in Multivariate Data Suitable for Use with an Electronic Computer
Authors: Buck S. F.
Published: December 2018

Estimation of statistical parameters from multivariate data results in wasted information, if units with incomplete data are rejected entirely, and perhaps in inconsistencies in the variance–covariance matrix if the variances and correlation coefficients are estimated from all available data on individual variates and pairs of variates respectively. An alternative is to estimate the missing values by regression techniques and to calculate a revised variance–covariance matrix. This method is suitable for use with an electronic computer. It is shown that with this method the resultant covariances are unbiased, but that the variances require correction for bias. A numerical example is given.
Evaluation of Determinants, Characteristic Equations and Their Roots for a Class of Patterned Matrices
Authors: Roy S. N. , Greenberg B. G. , Sarhan A. E.
Published: December 2018

The variance–covariance matrices arising in the analysis of variance, response-surface fitting and order statistics have special patterns expediting and facilitating their algebraic manipulation. By appropriate triangulation and partitioning devices, the determinants of these matrices can be evaluated from general formulae. The characteristic equations, and sometimes their roots, can also be obtained in a direct single step.
The Interaction Algorithm and Practical Fourier Analysis: An Addendum
Authors: Good I. J.
Published: December 2018

In the original paper it was misleadingly stated that the Yates algorithm is effectively self-inverse. This statement is here elucidated, and a numerical example is given. Attention is drawn to a discrete analogue of Poisson’s summation formula, which provides many additional relationships between yields and interactions.
Mixtures of Geometric Distributions
Authors: Daniels H. E.
Published: December 2018

The representation of a discrete distribution as a mixture of geometric distributions is discussed and applied to busy period distributions in equilibrium queueing systems.
Prediction by Exponentially Weighted Moving Averages and Related Methods
Authors: Cox D. R.
Published: December 2018

The mean square error of prediction is calculated for an exponentially weighted moving average (e.w.m.a.), when the series predicted is a Markov series, or a Markov series with superimposed error. The best choice of damping constant is given; the choice is not critical. There is a value of the Markov correlation ρ  0 below which it is impossible to predict, with an e.w.m.a., the local variations of the series. The mean square error of an e.w.m.a. is compared with the minimum possible value, namely that for the best linear predictor (Wiener). A modified e.w.m.a. is constructed having a mean square error approaching that of the Wiener predictor. This modification will be of value if the Markov correlation parameter is negative, and possibly also when the Markov parameter is near ρ  0.
The Intervals between Regular Events Displaced in Time by Independent Random Deviations of Large Dispersion
Authors: Lewis T.
Published: December 2018

Suppose events, planned as regular, in fact occur at instants displaced from the planned arithmetic progression by independent random deviations. Assume these deviations have a common continuous distribution, with variance so large that the realized order of the events differs from the planned order. The paper discusses statistical properties of the time intervals between such events, and compares some of the theoretical formulae with some known empirical results.
Exact Fiducial Limits in Non-Linear Estimation
Authors: Williams E. J.
Published: December 2018

In the problem of estimating an unknown parameter of a distribution (in normal regression or in more general situations), the determination of a point estimate and its distribution is contrasted with the setting of fiducial limits for the parameter value. The use of a point estimate is effective for setting fiducial limits only when the estimate is a sufficient statistic for the parameter. Exact fiducial limits can be determined for the non-linear parameters in a normal regression function. Let the regression of y on x be linear in f(x, γ), which is a non-linear function of γ. If f′(x, γ) is the derivative of f(x, γ) with respect to γ, the partial regression of y on f vanishes when γ takes the least-squares value. Exact fiducial limits are defined as those values making this partial regression coefficient significant at the required level of probability. A pathological example of fiducial limits is discussed. The extension of the principle to general estimation problems is indicated.
Queues in Series
Authors: Ghosal A.
Published: December 2018

It has been demonstrated in this paper how Lindley’s analytical method of solving the integral equation for the waiting-time distribution in a queue with a single server can be extended to obtain the waiting-time distribution in different queues, when queues are in series and each queue has a single server. The input has been taken as Poisson, but the service time has been assumed to follow a gamma distribution in the first queue and an exponential distribution in the second queue.
Some Optimal Sequential Schemes for Estimating the Mean of a Cumulative Normal Quantal Response Curve
Authors: Marks B. L.
Published: December 2018

If a relation between stimulus and response is quantal and the response curve is the cumulative of a normal distribution, then the mean of the underlying distribution can be estimated by performing trials with various stimuli. An estimation scheme is studied in which the stimuli are chosen in the light of the results of the previous trials and of the prior probability distribution for the mean. For some special cases optimal policies for choosing the stimuli are found analytically, or computed by dynamic programming, and compared with the optimal non-sequential layouts.
Further Results on Tests of Separate Families of Hypotheses
Authors: Cox D. R.
Published: December 2018

It is required to test a composite null hypothesis. High power is desired against a composite alternative hypothesis that is not in the same parametric family as the null hypothesis. In an earlier paper a modification of the Neyman-Pearson maximum-likelihood ratio test was suggested for this problem. The present paper gives some general comments on the formulation of the problem, a general large-sample form for the test, and, finally, a number of examples.
On the Consistency of the Fiducial Method
Authors: Fraser D. A. S.
Published: December 2018

Lindley (1958) and Sprott (1960) have proposed several criteria for the satisfactory application of the fiducial method of inference to statistical problems. The criteria are concerned with consistency when there are various ways of applying the method. In sections 1 and 2 these criteria are discussed briefly, and the range of problems that yield fulfilment of the criteria is shown to be considerably larger than that indicated by Lindley and Sprott. A frequency interpretation for a fiducial distribution has been proposed and discussed in two recent papers (Fraser, 1961a, b); this interpretation is available for problems that are invariant in a natural sense under translation and transformation groups. If the purpose of the fiducial method of inference is to produce posterior probability statements about the parameter that have a frequency interpretation and are consistent according to criteria such as Lindley’s and Sprott’s, then the range of validity can be substantially extended by making a small modification to the method; this is discussed in sections 3 to 6.
Estimation of Means and Standard Errors in the Analysis of Non-Orthogonal Experiments by Electronic Computer
Authors: Freeman G. H. , Jeffers J. N. R.
Published: December 2018

Various methods of estimating means and standard errors for non-orthogonal designs are considered from the point of view of their generality and suitability for programming on an electronic computer. The use of the variance-covariance matrix is shown to have a number of advantages, and this method is described for designs with two or three classifications; the method of analysis for three classifications, no two of which are orthogonal, is new. Programmes have been written for the Ferranti Pegasus computer for two and three classifications; these are outlined and an example of their use is given.
On the Comparison of the Sensitivities of Experiments
Authors: Dar S. N.
Published: December 2018

A large sample test for the comparison of the ratio of components of variance in two identical independent experiments is obtained by using a normalizing transformation. This test agrees closely with the exact test of Schumann and Bradley. Then large sample tests are developed for such comparisons in k identical experiments and for comparisons in dissimilar experiments.
A Comparison of the Direct and Fiducial Arguments in the Estimation of a Parameter
Authors: Williams E. J.
Published: December 2018

An approach to the problem of fiducial inference for dispersion on a sphere alternative to that adopted by R. A. Fisher is proposed.
On Direct Probabilities
Authors: Dempster A. P.
Published: December 2018

The concept of a direct probability argument is introduced. This is a means of assigning distributions a posteriori which satisfies the requirement that such distributions shall be valid conditional on the observations, but which differs from the inverse probability argument in that distributions a priori are not needed. Direct probabilities are closely related to fiducial probabilities. This theory requires a generalized concept of sampling a family Fθ of distributions which is discussed in Section 3. Direct inference from a sample of n is discussed in Section 4 for a single continuous parameter and in Section 5 for the non-parametric case.
The Logic of Least Squares
Authors: Barnard G. A.
Published: December 2018

Two alternative modes of developing the logic of the least squares theorem are presented. The first is based on the mean square error, the second on a notion of restricted sufficiency.
Models for a Bacterial Growth Process with Removals
Authors: Gani J.
Published: December 2018

The paper considers the extinction of a bacterial colony subject to phage infection. A description of the biological process leads to the construction of a simplified model from which it is possible to derive the probability generating function (p.g.f.) for surviving bacteria in terms of repeated contour integrals. The distributions of phage offspring released by a single bacterium, and of bacterial infections due to these phages, are discussed. For bacterial birth and birth-death processes, it is possible to express the p.g.f. for surviving bacteria explicitly; this is done by first obtaining the p.g.f. conditional on numbers of bacterial infections, and then summing over all such possible infections.
Some Remarks on a Method of Maximum-Likelihood Estimation Proposed by Richards
Authors: Kale B. K.
Published: December 2018

Richards (1961) suggested a method of maximum-likelihood estimation for the case of several parameters. Some gaps in the general theory given by him in support of the method are pointed out. Though the method is justifiable, the justification is shown to depend on the correct use of the implicit function theorem and is not connected with the joint asymptotic normality and the asymptotic efficiency of the maximum-likelihood estimates.
Hotelling’s Generalized T2 in the Multivariate Analysis of Variance
Authors: Ghosh M. N.
Published: December 2018

Hotelling’s generalized T2 is the sum of the roots of the determinantal equation |Z - θY/n| = 0, where Y and Z have central and non-central p-dimensional Wishart distributions with d.f. Q and n. Y and Z are obtained as the sum of products matrices for p-variate normal distributions N(0, σ1) and N(b(q), σ1). The distribution of this statistic is shown to be a monotonic increasing function of each of the population roots. An unbiased estimate of a linear function of the sum of population roots, which may be considered as a convenient measure of non-centrality, is also obtained from this statistic, and simultaneous confidence intervals for all linear functions of b(q) are obtained from it. The mean of the statistic is calculated for all p and the variance for p = 3 and p = 4.
On the Independence of Quadratic Expressions
Authors: Good I. J. , Welch L. R.
Published: December 2018

Let y1, y2, · · ·*** be pairwise independent quadratic expressions in x, not necessarily homogeneous, where x has a multinormal distribution. Then y1, y2, …*** are also mutually independent. Various conditions for independence are given. In the Appendix, it is shown that x is a linear transformation of a vector z, where z has a spherically symmetric distribution, and where the y’s depend on disjoint subsets of the components of z.
Periodogram Analysis and Variance Fluctuations
Authors: Herbst Laurence J.
Published: December 2018

Two problems in the subject of time-series analysis are considered: (A) the effect of fluctuating variances on the periodogram of a set of independent Gaussian data with zero means; (B) the possibility of using the periodogram of squares of independent Gaussian data with zero means as a tool in the Fourier analysis of the variances of the data. The basic effect of fluctuating variances on the periodogram is to introduce correlations (expressible in terms of the Fourier coefficients of the variance) which do not decrease as the sample size increases. The periodogram of squares of independent Gaussian data with zero means is shown to be an appropriate tool in the Fourier analysis of the variances.
Some Applications of Exponential Ordered Scores
Authors: Cox D. R.
Published: December 2018

There are simple standard tests for the comparison of samples assumed to arise from exponential distributions, but the properties of these tests are known to depend severely on the assumption of exponential form. However, the observations can sometimes usefully be ranked, and then replaced by the corresponding expected values of order statistics in sampling the unit exponential distribution, before calculating the appropriate test statistic. Some special tests of this type are examined. The procedure is analogous to the use of Fisher and Yates's scores in normal theory. Savage (1956) gave the test of this type for comparing two samples, in the course of a general study of rank tests for the two-sample problem.
A Note on Kurtosis
Authors: Finucan H. M.
Published: December 2018

This note “rediscovers” the original interpretation of kurtosis as an indicator of “a prominent peak and a prominent tail” on the density curve. A formal proof that such curves have indeed relatively high fourth moments is given. A comment is made concerning the recent decline of this correct classical interpretation.
A Deformation Method for Quadratic Programming
Authors: Zahl Samuel
Published: December 2018

This paper presents a new method for minimizing a positive definite quadratic function subject to linear inequality constraints. It is based on a continuous deformation of the quadratic, starting from one giving rise to an almost trivial problem and ending with the desired quadratic. The method is particularly suitable for problems (one such is outlined) involving large numbers of inequality constraints, since the size of the variable space is independent of the number of inequalities. The method is readily extended to linear programming by embedding the linear objective function in a second degree polynomial with positive definite quadratic component, solving the resulting problem by the above method, then driving the quadratic component to zero, i.e. deforming the second-degree programming problem back to the original linear one.
The Linear Hypothesis and Idempotent Matrices
Authors: Seber G. A. F.
Published: December 2018

The application of idempotent matrices to the testing of linear hypotheses is discussed and several properties of idempotent matrices are given to simplify the calculation of the F-test statistic. It is also shown that the F-test is based on the so-called Wald principle of large sample theory and this approach is put forward as an alternative to the usual likelihood ratio method of deriving the F-statistic.
Some Properties of Counts of Events for Certain Types of Point Process
Authors: Lewis T. , Govier L. J.
Published: December 2018

A stationary point process being supposed characterized by the counts Xl of events in time intervals of equal length l, a discussion is given of relationships, valid for a wide class of processes, between the variance-time function var Xl, the autocorrelation function of the Xl-series, and the autocovariance density. “Controlled variability” processes are then defined and discussed. A study is made of particular controlled variability processes, “(a, b) processes”, in which events occur at times a  0 + ia + bi, the bi being independent identically distributed random variables. In particular, it is shown that the autocovariance density for an (a, b) process is proportional to the convolution probability density of the difference of two independent bi's, with a closely related result for the autocorrelation function of counts.
A Branching Poisson Process Model for the Analysis of Computer Failure Patterns
Authors: Lewis Peter A. W.
Published: December 2018

Current reliability models predict that the failure pattern of a computer should form a Poisson process. Experience has shown, however, that the times-between-failures of a computer are not exponentially distributed, and that successive times-between-failures are not independent. These deviations from a Poisson process are due to imperfect repair of failed components; the resulting failure pattern is described by a branching Poisson process model. The probabilistic properties of this model are derived and used to analyse three series of computer failures.
Estimation in the Uniform Covariance Case
Authors: Geisser Seymour
Published: December 2018

There are three basic methods (confidence, fiducial, Bayesian) of obtaining interval estimates or estimating regions for the parameters of the multivariate normal distribution, none of which need necessarily yield a unique solution, although this has been claimed as a property of the fiducial argument. It has been previously demonstrated that they are not consistent with one another, i.e. they yield different regions for the same parameters. In this paper we show that if the structure of the covariance matrix is uniform the three methods yield the same regions for the vector mean and the same intervals for the common correlation coefficient.
The Determination of Sampling Distributions and Moment Generating Functions by Solving Differential Equations
Authors: Reitsma A.
Published: December 2018

A method of obtaining sampling distributions and moment generating functions by solving differential equations is introduced. This method is demonstrated in finding the distribution of the difference of two gamma variates. A method closely related to the above is used in obtaining the moment generating functions of certain Pearson-type distributions.
Supplement to “A Deformation Method for Quadratic Programming”
Authors: Zahl Samuel
Published: December 2018

This note has two purposes. The first is to show how additional linear inequality restrictions can be added to a quadratic programming problem, given the solution for the original set of inequalities, using the author's (1964) deformation algorithm. The second purpose is to replace the sufficient conditions of Proposition 2 of the earlier paper by a set of necessary and sufficient conditions which are easier to implement. These new conditions are based on the Kuhn and Tucker (1951, Section 3) necessary and sufficient conditions for an inequality constrained maximum. Because the new conditions are both necessary and sufficient, degeneracy three of the earlier paper cannot, in fact, occur.
Mixture Designs for Three Factors
Authors: Draper Norman R. , Lawrence Willard
Published: December 2018

Scheffé, in two recent papers (1958, 1963), has given designs for experimenting with mixtures. The basis of these designs is the choice of symmetrical arrangements of points in the factor space and the fitting of carefully chosen models which have exactly as many coefficients as there are data points. The designs are such that some of the experiments do not contain any of one or more ingredients of the mixture. This may or may not be a disadvantage depending on the problem involved. Here an alternative form of design selection is made for the case of mixtures of three factors. (The method has also been extended to four factors.) This involves the premise that, in the absence of specialist knowledge about the form of the true response function, it is desired to fit a response surface equation of first or second order over the factor space of possible mixtures, and experimental runs are needed which, in a certain sense, ensure the best surface fit possible. The principles used in the choice of appropriate designs will be those originally introduced by Box and Draper (1959).
Evolutionary Spectra and Non-Stationary Processes
Authors: Priestley M. B.
Published: December 2018

We develop an approach to the spectral analysis of non-stationary processes which is based on the concept of “evolutionary spectra”; that is, spectral functions which are time dependent, and have a physical interpretation as local energy distributions over frequency. It is shown that the notion of evolutionary spectra generalizes the usual definition of spectra for stationary processes, and that, under certain conditions, the evolutionary spectrum at each instant of time may be estimated from a single realization of a process. By such means it is possible to study processes with continuously changing “spectral patterns”.
Estimation by Ranking Parameters
Authors: Hartigan J. A.
Published: December 2018

It is shown that a linear ordering of the parameter space is a useful generalization of classical estimates, and that for a certain class of loss functions, Bayes decision procedures lead to very simple linear orderings.
Alternative Analyses of Contingency Tables
Authors: Gart John J.
Published: December 2018

Modifications of the logit and likelihood-ratio methods of analysing two-and three-dimensional contingency tables are developed from a semi-empirical approximation to the multinomial distribution. The modified likelihood-ratio test is shown to have an approximate relationship to the Freeman and Halton (1951) criterion for choosing a critical region for the exact test. A Bayesian argument leading to the same results is discussed.
Delay at Traffic Intersections
Authors: Hawkes A. G.
Published: December 2018

The distribution of delay to vehicles is found for two models of behaviour at traffic intersections. In the first part we consider vehicles turning both left and right from a minor road onto a major road which has two-way traffic. In Section 5 we study the delay on a major road caused by vehicles wishing to turn right and which must wait for a gap in the oncoming traffic in order to do so. Some numerical evaluation of mean delays shows the benefit which may be expected from certain road improvements.
A New Approach to Sampling from Finite Populations. I Sufficiency and Linear Estimation
Authors: Godambe V. P.
Published: December 2018

Barnard (1963) introduced the concept of linear sufficiency in connection with the Gauss-Markov set-up of estimation. In the present article the concept of linear sufficiency is redefined to suit the problem of estimation in survey sampling where, as demonstrated by the present author (Godambe, 1955, 1965), the Gauss-Markov set-up is fundamentally inapplicable. According to this redefined concept, a certain estimator is shown to be uniquely (up to a constant multiplier), linearly sufficient for the population total in the entire class (defined by the author (1955)) of linear estimators. The significance of this result will be clear on the background of the author's previous result (Godambe, 1955) demonstrating the non-existence of a uniformly minimum variance estimator in the entire class of unbiased linear estimators for the population total. Practically, the results are summarized in Theorems 5.1 and 6.3.
A New Approach to Sampling from Finite Populations. II Distribution-Free Sufficiency
Authors: Godambe V. P.
Published: December 2018

The idea that in some situations the prior knowledge about the unknown parameters could be formulated as a class of prior distributions, which could be used, not necessarily through Bayes posterior probability, for subsequent inference, is already present in Godambe (1955). In the present paper, the concept of distribution-free linear sufficiency or in short linear sufficiency, originally due to Barnard (1963b) but redefined by the present author (Part I), is extended by defining distribution-free sufficiency, removing the restriction of linearity. This extension again is based on the assumption that in some situations prior knowledge could be formulated as a class of prior distributions. A certain linear estimator of the population total, which in Part I was shown to satisfy the redefined criteria of linear sufficiency uniquely in the class of all linear estimators, is now shown to satisfy this extended criteria of distribution-free sufficiency in the entire class of estimators. Further the general relationship between the linear sufficiency of Part I and the distribution-free sufficiency introduced here is investigated. Broadly the result is that, if we restrict to linear estimators only, distribution-free sufficiency is identical with linear sufficiency. Finally, some remarks are offered by way of comparison between the result obtained by the author previously (Godambe, 1955) and the result here, about the utilization of the prior information. The approach of this paper clearly implies a generalization of Fisherian sufficiency suitable for the situations when prior knowledge consists of a class of prior distributions. An alternative generalization when the prior knowledge consists of a group structure is due to Barnard (1963a).
The Almost Full Dam with Poisson Input
Authors: Hasofer A. M.
Published: December 2018

In this paper, the content of a dam with infinite depth, unit release per unit time and inputs which form a compound Poisson process is investigated by means of an integral equation technique. The Laplace transform of the content distribution for an initially full dam is calculated and inverted. The asymptotic behaviour of the content is described and its mean and variance obtained. Duality relationships with the almost empty dam with Poisson input are pointed out and used to obtain various explicit expressions.
Some Asymptotically Efficient Sequential Procedures for Ranking and Slippage Problems
Authors: Srivastava M. S.
Published: December 2018

In this paper Chow and Robbins's (1965) sequential theory has been applied to ranking and slippage problems (including a multivariate slippage problem) and a class of asymptotically efficient sequential procedures is given for each problem. Except for the finiteness of the variance, no assumption regarding the form of the distribution function is made.
A Technique for Discussing the Passage Time Distribution for Stable Systems
Authors: Keilson Julian
Published: December 2018

For many processes arising in reliability and congestion studies, passage time distributions required are difficult to exhibit. An asymptotic discussion is often available centring about a limit theorem given previously, and asymptotic equality of the mean passage time and an associated mean sojourn time, the latter obtainable directly from the ergodic distribution for the process. Theorems are presented exhibiting this equality for a variety of Markov chains.
Topics in the Investigation of Linear Relations Fitted by the Method of Least Squares
Authors: Anscombe F. J.
Published: December 2018

Various topics are reviewed: the effect of modern computers on statistical calculation; analysis by vector components rather than analysis merely of variance; “stepwise regression”; testing goodness of fit by examining residuals. Possible alternatives to the method of least squares appropriate when the distribution of errors has long tails, and computation of the likelihood function resulting from a Pearson Type VII distribution of errors, are considered. Some of the methods discussed are illustrated by study of several sets of observations already well known in the literature.