bluuebunny/crossref_metadata_2025 huggingface dataset:
First-Passage Percolation
Authors: Hammersley J. M.
Published: December 2018

Morgan and Welsh (1965) discussed a two-dimensional Poisson growth process. The present paper generalizes their situation and proves some of their conjectures.
On the Distribution of the F-Type Statistics in the Analysis of a Group of Experiments
Authors: Caliński T.
Published: December 2018

Particular matrix representations are given for the usual sums of squares in the analysis of a group of experiments. Box's theorems on quadratic forms in normal variables are applied to study the distributions of the sums of squares and of the usual variance ratio statistics. The behaviour of these statistics under heterogeneity of the error variance and of the interaction variance is considered and approximate F-tests are suggested.
Some Aspects of Randomization
Authors: Box G. E. P. , Guttman Irwin
Published: December 2018

A key feature of Sir Ronald Fisher's work in designed experiments is randomization. The randomization schemes described by Fisher ensure that four important principles are met. Using these four principles as starting point, we investigate the requirements which these four conditions impose on the choice of design.
Optimum Allocation in Multivariate Surveys : An Analytical Solution
Authors: Kokan A. R. , Khan Sanaullah
Published: December 2018

Optimum allocation of the sample numbers when several characters are under study can be stated as a problem of non-linear programming whose solution can be attempted in various ways. In this paper we give an analytical solution to the allocation problem in a few important sampling procedures.
A Generalization of Fieller's Theorem to the Ratio of Complex Parameters
Authors: Halperin Max
Published: December 2018

By an elementary argument, essentially identical to that given by Fieller (1940) to obtain confidence limits for a ratio of real parameters, it is shown how one can obtain confidence regions for the real and imaginary parts of a ratio of complex parameters. An exact confidence region (under assumptions of normality) is developed based on Hotelling's T  2 distribution. A conservative region which allows a considerable simplification of computations is also suggested; this region is based on the joint distribution of two correlated Student t-variates.
Asymptotic Renewal Results for a Natural Generalization of Classical Renewal Theory
Authors: Heyde C. C.
Published: December 2018

A generalization of the classical renewal theory for non-negative summands is suggested which involves the maxima of the partial sums instead of the partial sums themselves. It is shown that the generalization is a natural one and that most of the classical properties are preserved. In addition, methods are developed for obtaining asymptotic expansions for the moments of the random variable whose expectation is the new renewal function.
Non-Homogeneous Branching Poisson Processes
Authors: Lewis Peter A. W.
Published: December 2018

The properties of non-homogeneous branching Poisson processes are derived, using some particular conditional distributional results for the non-homogeneous Poisson process. The properties derived include the Laplace-Stieltjes transforms of the cumulant generating functions of the number of events by time x and of the state of the process at time x.
On the Choice of a Strategy for the Ratio Method of Estimation
Authors: Rao T. J.
Published: December 2018

Three sampling strategies are considered: (i) H(M, R  1) consisting of the Midzuno (1952)–Sen (1952) sampling scheme and the estimator R1=X(∑i∈sYi)/(∑i∈sXi) of the population total Y, where the symbol ∑išs indicates that the summation is over all units Ui contained in the sample s; (ii) H(M, ŶHT) consisting of the Midzuno–Sen sampling scheme and the estimator ŶHT = ∑išs(Yi/πi), where πi is the probability of inclusion of the ith unit in the sample; (iii) H(πps, Rn) consisting of the πps sampling scheme (Hanurav, 1965; 1967) and the estimator Rn = (X/n) ∑išs(Yi/Xi) of Y. It is shown that the strategy H(πps, Rn) is suitable for the method of ratio estimation. A direct application to cluster sampling is given at the end.
The Combination of Information in Generally Balanced Designs
Authors: Nelder J. A.
Published: December 2018

A general method is proposed for combining information in generally balanced designs when estimates of treatment effects are available from more than one stratum. Estimates of stratum variances are obtained by equating residual sum of squares in each stratum to their expectations, using the ML estimates of treatment effects. Weights may be given, the solution then being explicit, or fitted, in which case iteration is required. The results of a Monte Carlo study are given for a BIB design to test the distribution of the variance estimates.
An Application of Biorthonormal Expansions in Theory of Stochastic Processes
Authors: Mehr C. B.
Published: December 2018

Markov processes (with the exception of Gauss-Markov processes) are not in general Markov in the wide sense. Using a biorthonormal expansion of the second-order density function, it is shown that a class of Markov processes (not necessarily Gaussian) are Markov in the wide sense. These processes share some of the nice properties of Gaussian processes.
Note on X2 Tests for Matched Samples
Authors: Bennett B. M.
Published: December 2018

This note represents an extension of the author's results (Bennett, 1967) on the problem of tests for matched samples. These were shown to be equivalent to tests of linear hypotheses H0 concerning multinomial probabilities. Solutions for the goodness-of-fit and modified minimum chi-square tests and an explicit relation between the two tests are presented. The power function is discussed under certain alternatives to H0.
On the Linear Control of a Linear System Having a Normal Stationary Stochastic Input
Authors: Dowson D. C.
Published: December 2018

This paper deals with the linear control of a linear system subject to a random error in the form of a stationary stochastic input. It is shown that when the error is normally distributed, the problem of finding stationary points of the expected value of almost any function of the input Yt and the output Xt is, in general, equivalent to that of finding stationary points of the quadratic control function Q = var Xt + φ cov (Xt, Yt) + θ var Yt, provided that θ and φ are suitably chosen. This leads to a procedure for finding the optimal linear control for all loss functions likely to be met in practice.
A Note on the Asymptotic Distribution of Sample Quantiles
Authors: Walker A. M.
Published: December 2018

This paper gives a rigorous proof, under conditions believed to be minimal, of the asymptotic normality of a finite set of quantiles of a random sample from an absolutely continuous distribution. The proof is substantially simpler than those that have previously been published.
Some Applications of Multiple-Type Branching Processes in Population Genetics
Authors: Ewens W. J.
Published: December 2018

Multiple-type branching processes are used in this paper to derive survival probabilities of mutant genes in a number of situations in population genetics. Specifically, we apply the theory in the following cases. (1) Bisexual populations. In a population allowing (say) far more males than females, a favourable new mutation will have far greater survival probability if it occurs initially in a female; use of weighted survival probabilities will allow derivation of an optimum sex ratio. (2) Two locus problems. The selective advantage of a gene is not a fixed quantity but depends on the “genetic background”, that is on the genes present at other loci in the individual in question. Here we suppose that the selective advantage depends, in this way, only on a second interacting locus. Attention will be concentrated on the effect of linkage between the two loci. (3) The extension to three locus problems. (4) In a model introduced by Kimura (1957) it is assumed that a chromosome consists of a number k of sub-units, or strands, in any one of which a mutation may occur. The mode of inheritance is such that each strand duplicates itself and the daughter chromosome is made up of a random collection of k of the 2k strands so obtained. This model allows a wide field of possible investigations, only a few of which are considered here.

Apart from questions of genetical interest, it is shown that a multiple-type branching process may not possess the close relationship with the corresponding deterministic process that is enjoyed by a single-type process. This means that some caution may be needed in drawing conclusions about the stochastic behaviour from the deterministic behaviour.
A Queue with Random Arrivals and Scheduled Bulk Departures
Authors: Mercer A.
Published: December 2018

The distributions of queue length and waiting time of a customer are considered for the queue where customers arrive at random, wait in the order of arrival and are scheduled to be served in batches by a single server during equal intervals. A general lateness distribution is assumed but a service either takes place in the scheduled interval or not at all; no overlapping of services is allowed.
The Bayesian Estimation of a Linear Functional Relationship
Authors: Lindley D. V. , El-Sayyad G. M.
Published: December 2018

The purpose of this note is twofold. First, to demonstrate by means of an example that a statistical approach based solely on the likelihood function can be misleading, whereas the Bayesian method yields sensible results. Second, to develop the Bayesian method and apply the results to the problem of estimating a linear functional relationship. In Section 1 we discuss the likelihood approach. Section 2 concerns the Bayesian method, which is applied to the functional relationship situation in Section 3: it is shown that the asymptotic behaviour of the distributions exhibits a novel form. In Section 4 the posterior distributions derived in Section 3 are considered in more detail: in particular, a standard error for the estimate of slope is provided. Section 5 contains various comments on the method, and, in particular, indicates other fields of application.
On Extremal Factorization and Recurrent Events
Authors: Heyde C. C.
Published: December 2018

Associated with any recurrent event there is a Markov chain whose distributions possess what can be called an extremal factorization property. The important role which this property plays in the general theory is briefly indicated and the behaviour of the probability of occurrence of the recurrent event, in some specified time interval, is discussed using the property as a tool.
Experiments with Mixtures: An Alternative to the Simplex-Lattice Design
Authors: Lambrakis D. P.
Published: December 2018

Scheffé (1958) introduced the simplex-lattice design for experiments with mixtures of q components. The purpose of this design is the empirical prediction of the response to any mixture of the components when the response depends only on the proportions of the components but not on the total amount of the mixture. In this paper an alternative to the simplex-lattice design is developed in which all the features of the design are maintained except that the pure mixtures are replaced by the (q − 1)-nary mixtures.
A Fiducial Argument with Application to Survey Sampling
Authors: Godambe V. P.
Published: December 2018

Sprott (1960) proposed certain restrictions on distributions a posteriori in terms of the likelihood function. Hacking (1965) put forward a criterion for irrelevance restricting the choice of the pivots in terms of the likelihoods. This paper presents an extended application of Hacking's criterion to accommodate ancillary statistics. This extension is utilized to obtain the fiducial distribution of the mean of a survey (finite) population based on the given random sample. The fiducial distribution so obtained is inseparably tied to the method of randomization and as such it provides an interpretation of “randomization”. In contrast, as shown by the author (1966a, b, c), the Bayes posterior of the population mean does not depend on “randomization”. This is further discussed in Section 8. An alternative method of obtaining the fiducial distribution via Bayes's theorem is explored in Section 9.
Two Queues in Parallel
Authors: Hunter Jeffrey J.
Published: December 2018

A queueing system consisting of two queues, each with single servers (independent negative exponential service time distributions) and correlated bivariate Poisson input, is discussed. The joint stationary probabilities for the respective numbers of customers in the two queues in equilibrium are found in the case where we have finite waiting rooms of different sizes for each individual queue. For the model where the waiting-room capacities are unlimited, we derive a functional relationship for the joint probability generating function of the equilibrium probabilities. A survey of published research on queues in parallel is also presented.
A Bivariate Non-Parametric c-Sample Test
Authors: Mardia K. V.
Published: December 2018

A non-parametric test for the bivariate c-sample problem is proposed. The test is an extension of a bivariate two-sample test given by Mardia (1967) and possesses various desirable properties. We derive the asymptotic distribution of this test and obtain the Pitman efficiency of the test relative to Wilks's criterion A for the translation types of alternatives. We give the critical values of the distribution and some approximations. The test is applied to a numerical example.
Durbin's New Multistage Variance Estimator
Authors: Brewer K. R. W. , Hanif M.
Published: December 2018

A new variance estimator was suggested implicitly by Durbin (1967), for use in a particular multistage sampling procedure. The suggestion can easily be extended to cover the general case in which selection at the first stage is without replacement (the probabilities of inclusion in sample being arbitrary) and in which unbiased estimators are used at all stages of sampling. As compared with the earlier Durbin multistage variance estimator, the new estimator has this advantage; that the conditional expectation (given any particular first-stage sample selection) of the contribution to the estimate of total variance from the second and lower stages is equal to the conditional contribution from these stages to the total variance itself. As a result of this advantage, the new estimator will nearly always be more efficient than the old. The results of a numerical example, however, suggest that the gains in efficiency are unlikely to assume large proportions, at least where the Horvitz–Thompson estimator of total is used. Formulae are also given for use with des Raj, Murthy, and Rao, Hartley and Cochran estimators.
Goodness of Fit of a Discriminant Function from the Vector Space of Dummy Variables
Authors: Kshirsagar A. M.
Published: December 2018

The problem of discrimination among several groups can be looked upon as the problem of “association” between the vector of variables under consideration and a vector of “dummy” variables representing the contrasts among the groups. In certain situations, observations on one or more of these dummy variables are explicitly available and tests have been derived by Bartlett and Williams, for the adequacy of a set of assigned discriminant functions from among these dummy variables. In this paper, the statistical distributions of these test criteria are derived analytically, assuming the null hypothesis to be true.
On a Statistic for Rank Analysis of Variance
Authors: Peritz E.
Published: December 2018

Gabriel and Sen (1968) have presented a family of statistics for univariate and multivariate one-way rank analysis of variance. The present note considers the univariate Gabriel–Sen statistic of the Mann–Whitney family. An elementary derivation of its asymptotic null distribution is pointed out. It is shown that simultaneous testing procedures (s.t.p.) based on the above statistic are asymptotically coherent.
Nearest-Neighbour Systems and the Auto-Logistic Model for Binary Data
Authors: Besag J. E.
Published: December 2018

Bartlett (1966) and Whittle (1963), respectively, have proposed alternative, non-equivalent definitions of nearest-neighbour systems. The former, conditional probability definition, whilst the more intuitively attractive, presents several basic problems, not least in the identification of available models. In this paper, conditional probability nearest-neighbour systems for interacting random variables on a two-dimensional rectangular lattice are examined. It is shown that, in the case of 0, 1 variables and a homogeneous system, the only possibility is a logistic-type model but in which the explanatory variables at a point are the surrounding array variables themselves. A spatial–temporal approach leading to the same model is also suggested. The final section deals with linear nearest-neighbour systems, especially for continuous variables. The results of the paper may easily be extended to three or more dimensions.
Results in the Theory and Construction of D-Optimum Experimental Designs
Authors: Wynn Henry P.
Published: December 2018

A method of generating D-optimum design measures as defined by Kiefer and Wolfowitz is briefly described. The relevant theorem is generalized to yield a method of generating D8-optimum design measures for a specific subset of s of the parameters in the regression model. The one parameter case is described in further detail. An analogous theorem for discrete designs with a fixed number of points is proved giving a lower bound to the optimality achievable with such designs.

A final section is devoted to the loss in efficiency entailed by almost equal replication of a given discrete design. The discussion is extended to the one-dimensional polynomial model and the optimality of the replication is conjectured for the general case and proved for the quadratic case.
A Combinatorial Proof of the Distribution of the Time to First Emptiness of an Infinite Dam with Markovian Inputs
Authors: Matthews Jane P.
Published: December 2018

In this note the combinatorial lemma of Mott (1963) for an infinite dam with independent inputs is extended to obtain the distribution of the time to first emptiness of an infinite dam with Markov dependent integral inputs subject to steady unit demand. A new expression for this distribution is obtained in terms of the trace of the matrix probability generating function of the inputs. The result is shown to agree with the expressions given by Gani (1969) and Brockwell and Gani (1970), in terms of the principal eigenvalue of this matrix.
Unbiasedness of Likelihood Ratio Confidence Sets in Cases Without Nuisance Parameters
Authors: Spjøtvoll Emil
Published: December 2018

Confidence sets based upon the likelihood function are constructed in the case without nuisance parameters. It is shown under certain assumptions on the density function that the confidence sets are unbiased. Various examples are given.
A Model for Studying Robustness of Estimators and Informativeness of Labels in Sampling with Varying Probabilities
Authors: Cassel Claes-Magnus , Särndal Carl-Erik
Published: December 2018

This paper presents a model for investigating how information contained in the labels of a finite population may be utilized in order to reduce variance when sampling from a finite population with varying probabilities. The model assumes a population of N values of a characteristic X given as Xi = Yi + Zi (i = 1, …, N), where the mean of X is to be estimated. Here Yi is a systematic component in the sense that its relation to the label i is a priori known, or intelligently guessed at, while Zi is an error component. In the with-replacement case, a sample of labels is selected with possibly unequal probabilities pi,(i = 1, …, N) attached to the various labels in each of the n draws. Assuming that N is reasonably large, the variance of the usual estimator is studied under various assumptions about the shape of the continuous distribution approximating the Y-distribution (e.g. gamma, Pareto, Weibull shapes), various assumptions about the influence of the error component Z, and different designs for distributing the probabilities Pi among the N labels.
An Asymptotically Efficient, Sequential Equivalent of the t-Test
Authors: Weiss L. , Wolfowitz J.
Published: December 2018

Let X1, X2, …, be independent, normally distributed chance variables, each with mean μ and variance σ2, both unknown to the statistician. Let it be required to test sequentially the hypothesis that (make a decision between) μ = μ1 against the alternative μ = μ2, μ1 < μ2. When the sample size is fixed the Student t-test is usually used for this purpose. In the present paper we give a sequential procedure for performing this test which is consistent and asymptotically efficient as (μ2 – μ1) approaches zero.
Central Limit Analogues for Markov Population Processes
Authors: McNeil Donald R. , Schach Siegfried
Published: December 2018

It is shown in this paper that several birth-death-type models (univariate as well as multivariate) can be approximated by Ornstein–Uhlenbeck processes as certain parameters become large. Such approximations are valuable whenever the Kolmogorov differential equation cannot be solved in closed form.
A New Test for Autocorrelated Errors in the Linear Regression Model
Authors: Berenblut I. I. , Webb G. I.
Published: December 2018

A test for autocorrelated errors in the linear model is introduced and shown to have, in general, greater power than the Durbin and Watson test for high values of autocorrelation.
Lagged Regression with Unknown Lags
Authors: Hannan E. J. , Robinson P. M.
Published: December 2018

The problem considered is that of determining the parameters in a lagged regression of one stationary time series on another when the lag is unknown and is not an integral multiple of the time interval between observations. The residual in the regression is also taken to be a stationary time series. The method of estimation is based on the maximization with respect to the lag of a form of autocorrelation between the two series. However, the autocorrelation is defined via the Fourier transformed data which enables non-integral lags to be considered and an optimal weighting of frequencies to be introduced. The estimation procedure's validity depends upon the satisfaction of an identification (aliasing) condition. Limit theorems are proved for the estimates and the method is extended to multivariate regressions. The method is applied to some economic and to some oceanographic data.
Bayesian Spline Regression when the Number of Knots is Unknown
Authors: Halpern Elkan F.
Published: December 2018

The problem of selecting the model and coefficients for a spline regression curve with normal errors is considered. The locations of all possible knots are assumed to be known; which subset of these is the set of actual knots is unknown. Each subset forms a model.

Priors of the form of a marginal distribution on the index of the model and a conditional distribution for each model on the coefficients and variance are allowed. The posterior distribution is explicitly derived for natural conjugate and vague priors.

The optimal predictor is derived for a loss function which is a generalization of that appearing in Lindley (1968). All that is required of the opinion is the marginal distribution for the model and the mean vector for the coefficients.
Multi-Factor Experiments in Non-Orthogonal Designs
Authors: Cotter Sarah C. , John J. A. , Smith T. M. F.
Published: December 2018

This paper generalizes the results of John and Smith (1972) to multi-factor designs.
Some Monotonicity Theorems for χ2, F and T Distributions with Applications
Authors: Ghosh B. K.
Published: December 2018

The probability that a non-central χ2, F or t variable exceeds c is a function of c, the degrees of freedom and the non-centrality parameter. The behaviour of these probabilities as functions of the degrees of freedom is investigated when c and the non-centrality parameters are fixed. It is also shown that, when c refers to the 100α% point of the corresponding central distribution, the probabilities are monotone in the degrees of freedom. Some allied results and possible applications are discussed.
Asymptotic Properties of the Wiener–Kolmogorov Predictor. I
Authors: Bhansali R. J.
Published: December 2018

“Windowed” estimates of the spectral density function are used to carry out the factorization of the sample spectral density function and hence estimate the Wiener-Kolmogorov prediction coefficients. Expressions are derived for the asymptotic mean and variance of all the statistics necessary to estimate the predictor. The covariances and asymptotic normality of these estimates are also investigated. In particular, an expression for the asymptotic mean-square error or predicting one step, and more than one step, ahead is given.
A Note on the Incompatibility of One-Sided Bayes and Frequency Confidence Intervals
Authors: Joshi V. M.
Published: December 2018

With a proper prior distribution the one-sided Bayes confidence interval for the scalar parameter θ in the density function f(x, θ) never coincides with the corresponding confidence interval based on frequency, whether the experiment is of fixed sample size or sequential.
Ridge-Type Estimators for Regression Analysis
Authors: Goldstein M. , Smith A. F. M.
Published: December 2018

An examination of the mean-square error properties of a class of shrinkage estimators for the normal regression model leads to a new derivation of the Hoerl–Kennard (1970) Ridge estimator and its generalization. Comparison is made with the James–Stein estimator, and with the generalized-inverse estimator proposed by Marquardt (1970). A Bayesian approach is noted.
Generalized Linear Models Specified in Terms of Constraints
Authors: Wedderburn R. W. M.
Published: December 2018

A modification of the method of Nelder and Wedderburn (1972) is given for fitting models with the same error distributions as discussed there but with the systematic part of the models specified in terms of constraints. It is possible to fit these by the method described by Nelder and Wedderburn using iterative weighted regression, but it turns out to be simpler to replace each regression calculation of that method by another one with the property that the fitted values of one regression calculation are the residuals of the other and vice versa. The method is applied to testing for marginal homogeneity in contingency tables.
Minimum Full Sequences of 2N–MResolution Iii Plans
Authors: Pajak Thomas F. , Addelman Sidney
Published: December 2018

A criterion is developed for determining the minimum number of stages required in order to obtain a sequence of 2n–m Resolution III plans which permits the estimation of all main and two-factor interaction effects. In order to achieve this minimum, equivalent families of defining contrasts are utilized to generate the treatment combinations which constitute each stage of the sequence.
The Influence of Reference Values and Estimated Variance on the Arl of Cusum Tests
Authors: Bagshaw Michael , Johnson Richard A.
Published: December 2018

An approximation, based on the Wiener process, is applied to Page's cusum statistic. Based on the average run length, the optimal reference value is determined analytically and shown to agree with results from studies using numerically determined run lengths. Next, we study the effect of using an estimate for variance and how this interacts with the choice of a reference value.
Comments and Queries on Patterns of Efficiency-Balanced Designs
Authors: Puri P. D. , Nigam A. K.
Published: December 2018

By an efficiency-balanced design is meant one in which all contrasts are estimated with equal efficiency. A sufficient condition is found in the off-diagonal elements of the matrix, nk−ş n', where n is the incidence matrix and k the vector of block sizes.
On Multi-Server Stochastic Scheduling
Authors: Glazebrook K. D. , Nash P.
Published: December 2018

Several identical components are available for use in a piece of machinery. The number of components needed to operate the machine is a function of time, probably constant, and the lifetime of each component is described by a probability distribution. After a certain number of component failures, there will not be enough components remaining to operate the machine. The problem is considered of how to use the available components to postpone this occurrence for as long as possible. The Lady's Nylon Stocking Problem of Cox is solved for certain classes of lifetime distribution.
A Bayesian Interpretation of Pretesting
Authors: Leamer Edward E. , Chamberlain Gary
Published: December 2018

The usual posterior mean or mode of a k-dimensional regression coefficient is shown to be (1) a weighted average of 2k constrained least-squares points, (2) a weighted average of k + 1 “ideal” points and (3) a weighted average of k + 1 principal component points. These results are used to interpret pretesting procedures which select a constrained least-squares estimate. For special classes of priors a somewhat distorted Bayesian analysis leads to a posterior measure of location that is an F-weighted average of restricted and unrestricted least squares. A spherical prior can be associated with methods that drop principal components. Multilevel testing is shown to imply a non-traditional prior.
The Subset Problem in Multivariate Analysis of Variance
Authors: Hawkins Douglas M.
Published: December 2018

As with most other multivariate techniques, methods such as discriminant analysis which depend on multivariate analysis of variance are affected adversely by the inclusion of irrelevant variables. This paper discusses the problem of finding a parsimonious subset of variables that is essentially as good as the full set. As a result of the close computational similarity to multiple regression, a stepwise procedure, and a best subset method are derived from their regression counterparts. The inferential aspects of the stepwise procedure are also investigated.
Resolvable Designs with Two Replications
Authors: Williams E. R. , Patterson H. D. , John J. A.
Published: December 2018

In this paper we show how the efficiency and optimality of two-replicate resolvable designs can be tested using a known relationship with symmetrical designs.
Non-Parametric Estimation of a Smooth Regression Function
Authors: Clark R. M.
Published: December 2018

This paper presents a new family of non-parametric estimators of a smooth regression function, which are shown to have theoretical advantages in small samples over some alternative estimators. The new estimator has practical computational advantages over spline functions, while its smoothing properties are very close to the optimal smoothing properties of cubic splines. A simulation experiment demonstrates the remarkable success of the cross-validation technique as a means of determining the appropriate degree of smoothing.
Spearman's Footrule as a Measure of Disarray
Authors: Diaconis Persi , Graham R. L.
Published: December 2018

Spearman's measure of disarray D is the sum of the absolute values of the difference between the ranks. We treat D as a metric on the set of permutations. The limiting mean, variance and normality are established. D is shown to be related to the metric I arising from Kendall's τ through the combinatorial inequality I ≤ D ≤ 2I.
A Note on Infinitely Divisible Point Processes
Authors: Shanbhag D. N. , Westcott M.
Published: December 2018

In the discussion of Cliff and Ord (1975), Westcott mentioned the problem of determining those point processes which are both doubly stochastic Poisson processes and Poisson cluster processes—the dual nature is of interest in model interpretation since the underlying mechanisms are radically different. This was related to the question of which doubly stochastic Poisson processes are infinitely divisible, and it was conjectured that this occurs if and only if the random rate process is infinitely divisible. The object of this note is to mention three relevant results.
Distributional Expansions for Canonical Correlations from Contingency Tables
Authors: O'Neill M. E.
Published: December 2018

The asymptotic distribution of the canonical correlations from contingency tables was investigated by the author (1978) for a general theoretical correlation structure. In this paper, the structure examined is one in which all theoretical, non-zero correlations are different. For the observed canonical correlations corresponding to non-zeros, an analytic expansion is obtained enabling not only the asymptotic distributions to be obtained, but also a correction for “bias”. A condition is found for determining when the sum of the canonical components corresponding to zero correlations is distributed as χ2.
An Optimal Strategy in Multi-Server Stochastic Scheduling
Authors: Weber Richard R. , Nash Peter
Published: December 2018

Identical components are available for use in a piece of machinery. The number of components needed to operate the machine is a function of time and the lifetime of each component is described by a known probability distribution. Once a certain number of components have failed there will not be enough left to operate the machine. We find a strategy which for certain lifetime distributions delays this occurrence for as long as possible.
A Unified Derivation of the Asymptotic Distributions of Goodness-Of-Fit Statistics for Autoregressive Time-Series Models
Authors: Hosking J. R. M.
Published: December 2018

The portmanteau and Quenouille goodness-of-fit tests for an ar(p) model are derived as likelihood-ratio tests against particular alternative hypotheses, and their asymptotic distributions are thereby obtained.
An Historical Note on Recursive Residuals
Authors: Farebrother R. W.
Published: December 2018

In this paper we establish that recursive residuals have been known since 1891.
On a Model for Concordance between Judges
Authors: Feigin Paul D. , Cohen Ayala
Published: December 2018

When a group of judges rank a collection of objects one might be interested in two questions : (i) What is the “true” or underlying order of the objects? and (ii) How homogeneously do the judges act? In this paper we investigate the ramifications, with respect to these questions, of a model originally proposed by Mallows (1957). It is shown that the model fits well to an empirical data set.
Cornish-Fisher Expansions for Confidence Limits
Authors: Winterbottom Alan
Published: December 2018

Cornish-Fisher expansions are adapted to provide approximate confidence limits for unknown parameters. Applications are made to the noncentral χ2 distribution, to the bivariate normal correlation coefficient and to maximum likelihood estimation.
The Estimation of an Autocorrelation Parameter of a Gaussian Vector Process
Authors: McDunnough Philip
Published: December 2018

A procedure based on differences is used to yield an estimator of an unknown parameter occurring in the autocorrelation structure of a stationary Gaussian vector process. Various properties of the estimator including its existence, consistency and asymptotic normality are shown. Necessary and sufficient conditions are given for the estimator to be efficient.
Maximum Likelihood Estimation of a Linear Structural Relationship with Replication
Authors: Chan Lai K. , Mak Tak K.
Published: December 2018

Maximum likelihood estimation of the parameters of a linear structural relationship when repeated observations are available is considered. The maximum likelihood estimate of the slope parameter is found to be a root of a fourth-degree polynomial and to be consistent as the number of replicates increases. The asymptotic variances and covariances of the estimates of the parameters are obtained through simplified formulae.
Edgeworth and Saddle-Point Approximations with Statistical Applications
Authors: Barndorff-Nielsen O. , Cox D. R.
Published: December 2018

A simple exposition is given of Edgeworth and saddle-point approximations for some univariate, multivariate and conditional distributions. The application of these approximations to some problems of conditional statistical inference within the exponential family is illustrated. Examples connected with the time-dependent Poisson process, the von Mises distribution and with bioassay are among those studied in a little detail. Some general results are derived about conditional likelihoods and about the distribution of the maximum likelihood ratio test statistic. An appendix discusses regularity conditions.
A Generalization of the Bayesian Steady Forecasting Model
Authors: Smith J. Q.
Published: December 2018

The Bayesian Steady Forecasting model is generalized to a very wide class of processes other than the normal by defining the time series on the decision space. Examples of such processes are presented including a Beta-Binomial process, a Poisson-Gamma process and a Student-t sample distribution steady model. Simple updating relations are given for most of the processes discussed.
Some Prior and Posterior Distributions in Survival Analysis: A Critical Insight on Relationships Derived from Cross-Sectional Data
Authors: Elandt-Johnson Regina C.
Published: December 2018

Many physiological characteristics such as, for example, blood pressure, level of serum cholesterol, vital capacity, may be considered as ageing factors contributing to the overall hazard rate function in an additive fashion (formula (3.2)). If so, the regression function of the characteristic on time, in a population of survivors to time t, depends on t even if the characteristic, itself, does not vary with t, for any one individual. Also, if there is a relationship between a characteristic and age for an individual, it cannot be uniquely estimated from population cross-sectional data, unless its functional contribution to the hazard rate is determined. Special consideration is given to the analysis, when the prior (i.e. at t = 0) distribution of each characteristic is normal, and the characteristics are independent.
Multi-Armed Bandits and the Gittins Index
Authors: Whittle P.
Published: December 2018

A plausible conjecture (C) has the implication that a relationship (12) holds between the maximal expected rewards for a multi-project process and for a one-project process (F and φi respectively), if the option of retirement with reward M is available. The validity of this relation and optimality of Gittins' index rule are verified simultaneously by dynamic programming methods. These results are partially extended to the case of so-called “bandit superprocesses”.
On the E-Optimality of Some Block Designs
Authors: Cheng Ching-Shui
Published: December 2018

This paper is concerned with the investigation of E-optimal block designs. Some conditions are given under which there is a regular graph design which is E-optimal. Immediate applications of these results demonstrate that a great majority of the designs obtained by John and Mitchell (1977) are indeed E-optimal. It can also be shown that a group-divisible design with λ1 = λ2 + 1 > 1 and group size 2 is E-optimal. The same result also holds for a PBIB design with a cyclic scheme, λ2 = λ1 ± 1, and v = 5. Another application shows that if d is a BIBD, a group-divisible design with λ2 = λ1 + 1, a group-divisible design with λ1 = λ2 + 1 > 1 and group size 2, or a PBIB design with a cyclic scheme, λ2 = λ1 ± 1 and v = 5, then the dual design of d is also E-optimal.
On the E-Optimality of Regular Graph Designs
Authors: Jacroux Mike
Published: December 2018

This paper is an investigation into the E-optimality of regular graph designs within various classes of proper block designs. Several sufficient conditions are given for the existence of an E-optimal regular graph design within the classes considered. It is shown that when a regular graph design exists whose minimum non-zero eigenvalue is large enough, then an E-optimal regular graph design exists. Several examples are given to show how the derived sufficient conditions can be applied.
Moments of the Least Squares Estimators in a Non-Linear Regression Model
Authors: Clarke G. P. Y.
Published: December 2018

The least squares estimates of the parameters of a non-linear regression model with normal and independent errors are expanded as a power series in independent standardized normal variables. From this expansion, expressions are derived for the variances and covariances which include one term beyond the usual approximation by the inverse of the information matrix. Similarly, expressions for the biases are found and overall relationships between the biases, variances and Beale's measures of non-linearity. Data following the Richards' Function are used in an example.
On Randomized Dynamic Allocation Indices for the Sequential Design of Experiments
Authors: Glazebrook K. D.
Published: December 2018

In response to a suggestion by Bather, Gittins proposed the adoption of rules for sampling one at a time from several Bernoulli populations which were based on randomized versions of his dynamic allocation indices. Such rules, Gittins conjectured, might sometimes be both asymptotically optimal and also (almost) Bayes with respect to the usual discounted reward criterion. This conjecture is investigated and verified.
The Geometry of Robust Procedures in Linear Models
Authors: McKean Joseph W. , Schrader Ronald M.
Published: December 2018

A geometry similar to the “co-ordinate-free” approach to least squares estimation in linear models is developed for R- and M-estimation. This is done by replacing Euclidean distance with appropriate distance measures. This approach allows development of a theory of estimable functions even in cases where expectations of errors do not exist, as well as a simple treatment of general linear hypotheses.
An Estimation Procedure of Parameters of a Certain Spectral Density Model
Authors: Taniguchi Masanobu
Published: December 2018

Let {X(t)} be a Gaussian stationary process with mean zero and true spectral density g(x). We shall fit a certain spectral model f  θ  (x) for this process, and propose an estimate of θ, say θ^n, by minimizing a criterion D{f  θ  , g), which measures the nearness of f  θ  (x) to g(x). Also we shall derive the asymptotic distribution of θ^n. For the case of some appropriate spectral parametrization of f  θ  (x) we show that θ^n can be expressed in an explicit form. Furthermore, we shall investigate the asymptotic efficiency of θ^n.
An Asymptotically Bayes Sequential Test for Comparing Two Normal Means
Authors: Havre L. S.
Published: December 2018

We consider sequential tests to decide whether the difference between the means of two normal populations is negative or positive. A Bayesian formulation is adopted, with costs of sampling which differ for the two populations and depend on the difference between the means. The asymptotic “shape”, in the sense of Schwarz (1962), of the Bayes continuation region is found, and this is used to derive an asymptotically Bayes test. A number of asymptotic optimality properties of this test are proved, and it is compared by computer simulation with a two-population SPRT and a truncated generalized two-population SPRT.
Revising Previsions: A Geometric Interpretation
Authors: Goldstein Michael
Published: December 2018

This paper considers the way in which an individual's previsions are changed by new information. Firstly, reasons are discussed for preferring prevision to probability as the fundamental element in the subjective formulation of statistical procedures. Then, following de Finetti, an inner product space is constructed, such that the vectors are random quantities of interest and the inner product corresponds to the previsions of the individual. The expected effect of new data is expressed by a bounded self-adjoint operator on this space. The information expected from the data is represented in terms of the eigenstructure of the operator. If the operator is compact, this representation has particularly simple form. New data impose orthogonal co-ordinate axes on the original inner-product space. The information expected about any random quantity is represented by a specified shrinkage along each axis. to any order of accuracy, all the information may be represented by a finite dimensional subspace. Finally, the geometric representation is related to more conventional, totally specified models.
A First Passage Problem in a Random Walk with a Quality Control Application
Authors: Munford A. G.
Published: December 2018

This paper is concerned with a first passage problem for a discrete time random walk which arises in a quality control scheme proposed by the author. There are two barriers, one of which is absorbing and the other such that the random walk particle is immediately returned to the origin upon reaching it. An explicit expression is derived for the probability distribution of the time to absorption when the barriers are symmetrically placed on either side of the origin.
Robust Sequential Approximate Bayesian Estimation
Authors: West M.
Published: December 2018

An approximation to the sequential updating of the distribution of location parameters of a linear time series model is developed for non-normal observations. The behaviour of the resulting non-linear recursive filtering algorithm is examined and shown to have certain desirable properties for a variety of non-normal error distributions. Illustrative examples are given and relationships with previous work on robustness and sequential estimation are mentioned.
A Hierarchical System of Interaction Measures for Multidimensional Contingency Tables
Authors: Agresti Alan
Published: December 2018

Two summary measures are proposed of the amount of rth degree (r + 1 factor) and higher interaction displayed in a (d + 1)-dimensional contingency table, 1 ≤ r ≤ d. The measures are based on a comparison of the “partially-raked” table having the r-factor interactions and lower-order effects removed to the “fully-raked” table in which all effects are removed. The measures have a proportional reduction in error construction with reference to using d of the variables to predict the remaining one within the partially-raked and fully-raked tables, and they are generalizations of the lambda and tau measures of association for cross-classifications of two nominal variables.
Prediction Models and Unequal Probability Sampling
Authors: Cumberland William G. , Royall Richard M.
Published: December 2018

Prediction models are used to study the Horvitz–Thompson estimator in probability–proportional-to-size sampling and six estimators of its variance. The variance estimators include the Horvitz–Thompson, Yates–Grundy, and jackknife statistics, as well as two bias-robust estimators from prediction theory. An empirical study confirms prediction theory's implications about these estimators.
A Credibility Interval for the Probability that a New Observation Belongs to One of Two Multivariate Normal Populations
Authors: Rigby R. A.
Published: December 2018

The moments of the logarithm of the likelihood ratio of a new observation, drawn from one of two multivariate normal populations, are obtained from the posterior moment generating function. The distribution of the logarithm of the likelihood ratio is approximated using Pearson curves. The distribution of the probability p that a new observation belongs to one of the two populations can be obtained by transformation. The relationship between the expected value of this distribution and the “predictive” estimate of p is shown.
Conjectures for Optimal Block Designs
Authors: John J. A. , Williams E. R.
Published: December 2018

In the study and construction of optimal block designs over the past 20 years a number of conjectures have been repeatedly stated and used. The purpose of this note is to bring together some of these conjectures. Although known to those actively engaged in research in design, it is hoped that this note will stimulate other statisticians and mathematicians to give their attention to these unsolved problems.
Testing Separate Families of Hypotheses: An Information Criterion
Authors: Sawyer K. R.
Published: December 2018

An information criterion is developed for the test of a composite null hypothesis against a composite separate alternative. The intuitive and interpretative appeal of the criterion is emphasized and comparisons with the Cox test invoked.