bluuebunny/crossref_metadata_2025 huggingface dataset:
Linear Approximation Using the Criterion of Least Total Deviations
Authors: Davies M.
Published: December 2018

A study is made of the fitting of discrete data by linear regression planes and by polynomials according to the criterion of least total absolute deviations. A form of solution is proposed which makes use of the simplex method of linear programming. An example is given to show the practical application of the method.
Ex Post Determination of Significance in Multivariate Regression When the Independent Variables are Orthogonal
Authors: Geary R. C.
Published: December 2018

In multivariate regression, when the regressors are orthogonal, the estimates of the coefficients may be regarded, in the normal case, as an independent normal random sample with estimable variance. Significance is determined by the absolute magnitude of the highest member of the sample, after consideration of order statistics more generally in this context.

The method is applied to time series data analysed by Fisher and Yates (1957): while these authors identified, by their essentially ex ante approach, the first and second orthopolynomials as significant, ex post only the second is identifiable.
The Analysis of Association Among Many Variates
Authors: Williams E. J.
Published: December 2018

The methods used to estimate the nature and form of linear relations among sets of variates are developed in a simple manner based on a multivariate extension of the analysis of covariance. The link between these methods of analysis and the canonical analysis is discussed; it is shown how the two analyses can complement each other. Since it is of practical importance to represent the association among the variates in terms of a minimal set of derived variates, various tests of significance for the adequacy of such sets are presented. The methods make no use of non-central distributions, the only distribution theory involved being that of determinantal ratios whose three parameters are effectively the numbers of variates in the two associated sets and the number of observations of each. The methods have been set out in enough detail to show that they are not only relevant to many practical situations but also readily computable.
The Theory of Risk
Authors: Borch Karl
Published: December 2018

The classical theory of risk is reviewed. It is shown to be a static theory and this is adduced to be its main disadvantage. A dynamic theory was introduced by Lundberg in 1909. This is discussed and Lundberg's achievement is reaffirmed. However, it is pointed out that this theory, in spite of extensive mathematical development, has found virtually no application in practice. The major difficulty lies in the formulation of insurance problems in a convincing way. In the last section a new approach is called for. A model incorporating dynamic features of Lunberg's collective risk theory with more realistic constraints is investigated. Possible directions for future research are suggested.
An Analysis of Departures from the Exponential Distribution
Authors: Jackson O. A. Y.
Published: December 2018

A test for a Poisson process based on ordered sample values is proposed and its distribution and other properties derived. The test statistic is shown to be asymptotically normal and some further approximations to its distribution are investigated. The asymptotic relative efficiency of the test with respect to the asymptotically most powerful test against the alternative of gamma-distributed intervals is derived. Finally, a test for an incomplete sample is outlined.
A Note on a Modified Exponentially Weighted Predictor
Authors: Blight B. J. N.
Published: December 2018

A generalized form of a modified exponentially weighted predictor is considered and, for a Markov process, it is optimized with respect to the modification.
An Estimation Procedure for Mixtures of Distributions
Authors: Choi Keewhan , Bulgren W. G.
Published: December 2018

Notation

The symbols for estimators in this paper have been printed in light italic type. This is an exception from our usual typographic convention of showing matrices and vector quantities in bold black type, since in this case the large number of such symbols would seriously interfere with readability.

A general method is proposed for estimating the mixing measures of combinations of known distributions. The method is applicable to continuous and discrete mixtures. The small sample behaviour of the method is investigated by Monte Carlo methods. In large samples it is shown that the estimator converges to the true mixing measure with probability 1 at the rate of almost O(n-½) and has an asymptotic normal distribution.
Multivariate Beta Distribution and a Test for Multivariate Normality
Authors: Wagle B.
Published: December 2018

This paper presents an application of the multivariate beta distribution to the problem of testing whether a given sample of multivariate observations could have come from a multivariate normal population with an unknown mean vector and dispersion matrix. By using a series of transformations and the principle of randomization it is shown that the above problem can be reduced to that of testing for univariate normality. No attempt is made to investigate the power of this test; but some brief thoughts are expressed in the concluding section.
On the Concept of the Spectrum for Non-Stationary Processes
Authors: Loynes R. M.
Published: December 2018

Following a brief general description of the attitude adopted here to statistical models and definitions, a study of the problem of defining a spectrum for non-stationary processes is made. The approach is to determine first what one would wish to use such a spectrum for, and then attempt to discover whether this is possible. By the nature of the case no conclusive answer is possible, but the tentative conclusion drawn is that it is not.
Estimation of the Cross-Spectrum of a Stationary Bivariate Gaussian Process from its Zeros
Authors: Brillinger David R.
Published: December 2018

Under regularity conditions the cross-spectrum of a stationary, bivariate, Gaussian process, continuous in mean and with unit variance, may be estimated consistently as T → ∞ from the zeros in the interval [0, T] of its two components and the knowledge that the components have the same or opposite sign at time 0. The power spectra of the components may be estimated from the respective components' zeros alone. The estimates are asymptotically normal and unbiased. During the course of the paper a bound, that may be of independent interest, is given for the joint cumulants of certain functions of multivariate normal random variables. A general theorem on the joint asymptotic normality of spectral and cross-spectral estimates is presented. There is also a worked example.
The Stationary Distribution of a Branching Process Allowing Immigration: A Remark on the Critical Case
Authors: Seneta E.
Published: December 2018

For a branching process in which the individuals reproduce from generation to generation as in a Galton-Watson process, and are also subject to an immigration component at each generation, it is known that a stationary-limiting distribution sometimes exists. In particular, this is so if the mean number of offspring per individual is less than unity and the immigration distribution is reasonably behaved. We show that such a distribution may exist even when the mean number of offspring is unity, under special circumstances for which we give sufficient conditions.
Stochastic Models of Capital Investment
Authors: Plackett R. L.
Published: December 2018

An asset generates profits of random amounts at random points in time. We discuss the distribution theory of total profits for a given period under discounting and reinvestment. The problem of replacement is also considered. An application of the results is made to Premium Savings Bonds.
On the Null Distribution of a Non-Parametric Test for the Bivariate Two-Sample Problem
Authors: Mardia K. V.
Published: December 2018

A non-parametric test criterion U for the bivariate two-sample problem has been proposed by Mardia (1967). It is shown in Mardia (1968) that a two-sample test on a circle proposed by Wheeler and Watson (1964) has the same null distribution as that of U. In view of this additional application, we extend the table of critical values of U given in Mardia (1967), give a new approximation to the null distribution and compare it with the approximations given in Mardia (1967). We also apply the U-test to a geological problem.
Regression Problems When the Predictor Variables are Proportions
Authors: Becker N. G.
Published: December 2018

That proportions must sum to unity results in the need for adaptations to commonly used regression techniques when these are applied to proportions. Some such adaptations are considered here.
Estimating the Parameters of a Convolution
Authors: Sclove S. L. , van Ryzin J.
Published: December 2018

Parametric estimation problems are considered when one observes a random variable which is the sum of two independent non-identically distributed random variables. Some basic identifiability theorems are given which suggest the method of moments as a general method for attacking such problems, particularly when one of the summand variables is discrete. Moment estimators and their asymptotic distributions are examined for various specific examples. Some applications are suggested.
Inference Concerning Probabilities and Quantiles
Authors: Green J. R.
Published: December 2018

It is intended to make inferences on the basis of a set of observations resulting from n replicates of an experiment, where each observation X has distribution F(x; θ), in order to provide information about a future performance of the same experiment, or a small number of such future performances. In such a situation the inference should concern the probability of an event occurring in the future experiment, rather than directly concern the parameter θ (a vector in general). Appropriate kinds of inference are considered, using the correspondence between tolerance limits and tests and confidence intervals for quantiles. The interpretation of tolerance intervals, etc. is considered.
Tests for Specification Errors in Classical Linear Least-Squares Regression Analysis
Authors: Ramsey J. B.
Published: December 2018

The effects on the distribution of least-squares residuals of a series of model mis-specifications are considered. It is shown that for a variety of specification errors the distributions of the least-squares residuals are normal, but with non-zero means. An alternative predictor of the disturbance vector is used in developing four procedures for testing for the presence of specification error. The specification errors considered are omitted variables, incorrect functional form, simultaneous equation problems and heteroskedasticity.
Distinguishing Among Multinomial Populations: A Decision Theoretic Approach
Authors: Stromberg John L.
Published: December 2018

This paper presents a method for determining, on the basis of sample data, which of several multinomial populations are the same and which are distinct. The method is compared with alternative approaches and is shown to have certain advantages. its limitations are also discussed. It is then applied to the specific anthropological problem of distinguishing among six burial sites of Northern California Indians.
Resolution Iv Fractional Factorial Designs
Authors: Margolin Barry H.
Published: December 2018

Fractional factorial designs of resolution IV permit estimation of all the main effects with no aliasing by two-factor interactions. This paper produces a lower bound for the number of observations required for a general fractional factorial design to be of resolution IV. This lower bound agrees with a lower bound obtained by Rao for orthogonal arrays of strength 3. In addition, it is proved that this lower bound is attainable for the t. 2n factorial design series for t even and n ≡ 3 (mod 4) in plans which permit orthogonal estimation of the main effects. Finally, Webb's conjecture that there exist no resolution IV 2n factorial designs with 2n runs except for those constructed by the fold-over technique is proved to be valid.
Multicollinearity and Imprecise Estimation
Authors: Silvey S. D.
Published: December 2018

For the standard linear model containing several explanatory variables, the precision of estimation of linear parametric functions is analysed in terms of latent roots and vectors of X'X, where X is the matrix of values of explanatory variables. This analysis provides a practical method for detecting multicollinearity, and it is demonstrated that it is also useful in solving problems of optimum choice of new values of explanatory variables.
On Priority Queues in Heavy Traffic
Authors: Mazumdar Sati
Published: December 2018

For single-server, first-come-first-served queues, specified completely by the sequences of interarrival times and service times, if these sequences are stationary and if (in some sense) there is not too much dependence within or between them, it has been shown by Kingman (1962) that the waiting-time distribution is, in heavy traffic, approximately negative exponential. This result has been extended in the present paper when there are two types of customer, one type of customer having priority of service over that of the other.
On the Asymptotic Relative Efficiencies of Certain Location Parameter Estimates
Authors: Loynes R. M.
Published: December 2018

If, by analogy with means and medians, one computes that value such that the sum of the absolute pth powers of the sample deviations about that point takes on its minimum, one obtains an estimate of the central value of a symmetric distribution. Such estimates, for p ≥ 1 and under reasonable conditions, are known to be asymptotically normal as the sample size becomes large, and here the asymptotic relative efficiency of any pair is obtained, both for the general case and when the distribution sampled is also supposed unimodal.
On a Sequential Analogue of the Behrens–Fisher Problem
Authors: Srivastava M. S.
Published: December 2018

Srivastava (1966) proposed a class of asymptotically efficient sequential procedures for an analogue of the Behrens–Fisher problem. Recently, Robbins et al. (1967) also proposed the same class of sequential procedures. The object of this note is to give the details of our proof for the asymptotic efficiency, assuming only the finiteness of the variances.

The multivariate analogue of this problem has also been considered.
Estimation of the Branch Points of a Branching Diffusion Process
Authors: Edwards A. W. F.
Published: December 2018

A solution to the problem of estimating the positions and times of the branch points of a Brownian-motion/Yule process, given the positions of all the particles at a particular time, is outlined. A likelihood approach is used, and it is shown that the solution involves maintaining a clear distinction between likelihood and conditional probability if difficulties over mathematical singularities are to be avoided. Several unsolved mathematical problems are encountered, and it is concluded that some simulation studies may be required for a complete solution. Questions of scientific inference which the problem raises are discussed briefly.
A Short Cut in the Fitting of Autoregressive Series
Authors: Ray W. D.
Published: December 2018

The set of k equations arising from the asymptotic least-squares estimation procedure for fitting the unknown parameters in kth-order autoregressive models is shown to degenerate into two subsets, each of m equations for the even case (k = 2m), or two subsets, one of m equations and one of (m + 1) equations for the odd case (k = 2m + 1).
Simultaneous Confidence Bands in Regression Analysis
Authors: Wynn Henry P. , Bloomfield P.
Published: December 2018

Simultaneous confidence intervals for linear combinations of parameters in a regression analysis are discussed, and a desirable property is described. Confidence intervals having this property are characterized and the Working–Hotelling band is discussed with reference to it. It is shown how this band may be sharpened to give exact confidence levels in linear and quadratic regression. Tables are given from which the confidence bands may be constructed.
Power Families: A Note on the Box and Cox Transformation
Authors: Schlesselman J.
Published: December 2018

The family of power transformations defined by Box and Cox (1964) is shown to lead to likelihood procedures which are scale invariant only if the design matrix allows for removal of an additive constant. Modifications of the transformation family are suggested which preserve both continuity in the transformation and scale invariance in the likelihood procedures.
The Effect of the Demand Distribution in Inventory Models Combining Holding, Stockout and Re-Order Costs
Authors: Foster F. G. , Rosenhead J. V. , Siskind V.
Published: December 2018

The well-known square-root formula of inventory theory is based on the assumption of known, constant demand. When demand is probabilistic, the values of inventory control variables which minimize the expected total costs will depend on the shape of the distribution. In this paper we consider the effect of several demand distributions on a re-order cycle inventory system; however, the method of analysis is standard and could be applied to a number of other models. Formulae for the iterative calculation of optimal values of control variables are derived, and illustrated by worked examples. In the process, some results in distribution theory are developed.
Bayes, Fiducial and Frequency Aspects of Statistical Inference in Regression Analysis in Survey-Sampling
Authors: Godambe V. P. , Thompson Mary E.
Published: December 2018

In this paper an attempt is made to show how the regression analysis generally used for hypothetical populations can also be validated for the actual populations commonly dealt with in statistical surveys. This validation is based on a frequency distribution generated by randomization. In this context some comments are made on the relationships between Bayes, fiducial and frequency theories of statistical inference.
Point Spectra of Some Mutually Exciting Point Processes
Authors: Hawkes Alan G.
Published: December 2018

The point spectral matrix is obtained for a class of mutually exciting point processes. The solution makes use of methods similar to those used in solving Wiener–Hopf integral equations.
A Multi-Sample Uniform Scores Test on a Circle and its Parametric Competitor
Authors: Mardia K. V.
Published: December 2018

We propose a c-sample non-parametric test W on a circle based on uniform scores. The test can be regarded as an extension of Wheeler and Watson's test (1964). This test is constructed with the help of a variant of a c-sample test for von Mises populations under the shift-type of alternatives. We indicate the relationship of the W-test with a test given by Mardia (1970). This result solves the problem of critical values of this test and of approximations to the null distribution of W. The asymptotic distribution of this test and of its parametric competitor test S are obtained. Bahadur's efficiency of the W-test relative to the S-test is derived and some important particular cases are considered. The tests are applied to a numerical example.
A Bivariate Exponential Distribution with Applications to Reliability
Authors: Hawkes Alan G.
Published: December 2018

In a recent paper Downton introduced a bivariate exponential distribution in a reliability context. This note proposes a more general distribution which may have some advantages in practice.
Convex Loss Applied to Design in Regression Problems
Authors: Laycock P. J.
Published: December 2018

A general linear regression function is to be observed at n points in order to estimate a known linear combination of the unknown parameters. The n points and the estimator are to be optimum in some sense and in this paper the main criterion for optimality involves uniformly minimizing certain convex loss functions. Three main sets of results are obtained, followed by some further results for normally distributed errors.
Non-Parametric Function Fitting
Authors: Priestley M. B. , Chao M. T.
Published: December 2018

In this note we consider the problem of fitting a general functional relationship between two variables. We require only that the function to be fitted is, in some sense, “smooth”, and do not assume that it has a known mathematical form involving only a finite number of unknown parameters.
Some Power Comparisons between Bivariate Tests and a Twofold Univariate Test
Authors: Genizi Abraham
Published: December 2018

The paper describes a comparative study of the power of four bivariate tests of linear hypotheses and of a “twofold univariate” test equivalent to applying separate F-tests to the two variables. The powers of the multivariate tests are estimated by Monte Carlo methods, and that of the twofold univariate test is derived by a method using characteristic functions. Graphs describing the powers are given, leading to practical recommendations in terms of parameters which have a simple intuitive meaning.
Mean-Value Formulae for a Class of Random Sets
Authors: Streit F.
Published: December 2018

Mean-value formulae for important characteristics (e.g. volume, surface area, etc.) of the common intersection of randomly placed point-sets are derived under the condition that the basic probability distribution of the underlying model is either motion-invariant or translation-invariant (the orientation of the generating sets in this latter case being known). Various special cases are discussed, some of them leading to a new derivation of known results of geometrical probability theory. The expected value of the probability that an arbitrary point is covered by the random intersection-set is also determined.
A Test of Fit for Bivariate Distributions
Authors: Dahiya Ram C. , Gurland John
Published: December 2018

Tests of fit based on generalized minimum χ2 techniques are developed for bivariate distributions. The asymptotic null distribution of the test statistic is χ2 while the asymptotic non-null distribution turns out to be that of a weighted sum of independent non-central χ2 variates. The special case of testing the fit of a bivariate normal distribution is investigated in detail and the power is obtained for several alternative families of bivariate distributions.
Scale Mixtures of Normal Distributions
Authors: Andrews D. F. , Mallows C. L.
Published: December 2018

This paper presents necessary and sufficient conditions under which a random variable X may be generated as the ratio Z/V where Z and V are independent and Z has a standard normal distribution. This representation is useful in Monte Carlo calculations. It is established that when 1/2V  2 is exponential, X is double exponential; and that when 1/2V has the asymptotic distribution of the Kolmogorov distance statistic, X is logistic.
Partial Confounding in Symmetrical Factorial Experiments
Authors: Cotter Sarah C.
Published: December 2018

A new method is given for partial confounding with any block size in symmetrical factorial experiments. The method combines the techniques of confounding and incomplete block design, so that some factorial effects are completely confounded and some are partially confounded. A list of useful balanced incomplete block designs and group divisible designs is given, showing the wide range of available block sizes.
Single Replicate Factorial Experiments in Generalized Cyclic Designs: II. Asymmetrical Arrangements
Authors: Dean A. M. , John J. A.
Published: December 2018

This paper extends the results of John and Dean (1975) to asymmetrical factorial experiments, and lists a number of designs.
Uniqueness Relations for Linear Posterior Expectations
Authors: Goldstein M.
Published: December 2018

When the posterior expectation of the population mean is a linear function of the sample observation, then the moments of the prior distribution are uniquely determined. From this, a fairly general condition is given for establishing that the prior distribution for the population mean has a unique form.
A Note on the Maximum Size of a Closed Epidemic
Authors: Barbour Andrew D.
Published: December 2018

Daniels successfully approximated the distribution of the maximum number of infectives during an epidemic in a closed population, by reducing the problem to a curved boundary problem for Brownian motion. The purpose of this note is to present a simpler way of tackling the latter problem, using the invariance properties of Brownian motion.
A Test for Normality Based on Sample Entropy
Authors: Vasicek Oldrich
Published: December 2018

A test of the composite hypothesis of normality is introduced. The test is based on the property of the normal distribution that its entropy exceeds that of any other distribution with a density that has the same variance. The test statistic is based on a class of estimators of entropy constructed here. The test is shown to be a consistent test of the null hypothesis for all alternatives without a singular continuous part. The power of the test is estimated against several alternatives. It is observed that the test compares favourably with other tests for normality.
Estimation of the Second-Order Intensities of a Bivariate Stationary Point Process
Authors: Brillinger David R.
Published: December 2018

We consider histogram and smoothed histogram type estimates of the auto and cross intensity functions of a bivariate stationary point process. The asymptotic distributions are found to be multiples of Poissons in the histogram case and linear combinations of Poissons in the smoothed case. These asymptotic distributions suggest the plotting of the square roots of the estimates in order to stabilize the variance and to make the distributions more nearly normal. Two examples of such plots are presented in the paper.
Forecasting Transformed Series
Authors: Granger C. W. J. , Newbold P.
Published: December 2018

Suppose that a forecasting model is available for the process Xt but that interest centres on the instantaneous transformation Yt = T(Xt). On the assumption that Xt is Gaussian and stationary, or can be reduced to stationarity by differencing, this paper examines the autocovariance structure of and methods for forecasting the transformed series. The development employs the Hermite polynomial expansion, thus allowing results to be derived for a very general class of instantaneous transformations.
Sampling Theory for Opaque Spatial Specimens
Authors: Davy Pamela , Miles R. E.
Published: December 2018

Procedures for inferring the internal geometric structure of an opaque bounded n-dimensional set by random sectioning are discussed. General conditions are given (evidently for the first time) for the validity of the so-called “fundamental formulae” of stereology (the science of inferring spatial structure from lower dimensional data). In technical terms, a wide class of unbiased ratio estimators, based on the traces of embedded features in appropriately weighted random sections, is presented.
Modelling Spatial Patterns
Authors: Ripley B. D.
Published: December 2018

Spatial point processes may be analysed at two levels. Quadrat and distance methods were designed for the sampling of a population in the field. In this paper we consider those situations in which a map of a spatial pattern has been produced at some cost and we wish to extract the maximum possible information. We review the stochastic models which have been proposed for spatial point patterns and discuss methods by which the fit of such a model can be tested. Certain models are shown to be the equilibrium distributions of spatial–temporal stochastic processes. The theory is illustrated by several case studies.
Marginal Likelihood Solution to Some Problems Connected with Regression Analysis
Authors: El-Shaarawi A.
Published: December 2018

In the linear regression model when the design matrix x is a function of a set of unknowns ρ, a marginal likelihood is derived for making inferences about p. Several special cases are considered and a numerical illustration is given.
A Large-Sample Test for Detecting Gaps in Moving Average Models
Authors: Godolphin E. J.
Published: December 2018

A procedure is given in the paper for testing the hypothesis that a specified set of parameters in a moving average process assumes the value zero. In particular, the method is applicable for detecting whether a moving average process is seasonal or not. This large-sample test, which operates entirely in the time domain, is based on the estimated asymptotic covariance matrix of improved estimates of the autocorrelation function and is derived from the results of Walker (1961) and Godolphin (1977). The procedure is illustrated and compared favourably on the basis of test power with the identification approach of Box and Jenkins (1970).
A Scale-Free Goodness-Of-Fit Test for the Exponential Distribution Based on the Gini Statistic
Authors: Gail M. H. , Gastwirth J. L.
Published: December 2018

The sample Gini index is shown to be a powerful scale-free test of exponentiality against a variety of alternatives. Compared to the maximum likelihood test, the asymptotic relative efficiency of the Gini statistic is 0·69 against gamma and 0·88 against Weibull alternatives. Exact and asymptotic distribution theory is developed. On the basis of good power compared to competing tests, ease of computation, availability of exact critical values and robustness to measurement error, we recommend the Gini statistic as a scale-free goodness-of-fit test for the exponential distribution.
Remarks on a Paper by Cornfield and Detre
Authors: Kalbfleisch J. D. , MacKay R. J.
Published: December 2018

The method of non-parametric Bayesian analysis of continuous life table data proposed by Cornfield and Detre (1977) is shown to be incorrect. A more careful analysis leads to the results of Kalbfleisch (1978).
On Parameter Estimation for Spatial Point Processes
Authors: Diggle Peter J.
Published: December 2018

In a recent paper, Ripley (1977) discusses the use of two goodness-of-fit criteria for spatial point pattern data. Both criteria can be adapted to the associated problem of parameter estimation. to illustrate their use in this respect, we re-work one of Ripley's examples and reach rather different conclusions.
A Model for Cylindrical Variables with Applications
Authors: Mardia K. V. , Sutton T. W.
Published: December 2018

A suitable model when two random variables take values on a cylinder is proposed and its main properties are discussed. The maximum likelihood estimators for the parameters of the model are obtained and a likelihood ratio test of independence is derived. A numerical example is given.
Studies in the Robustness of Multidimensional Scaling: Procrustes Statistics
Authors: Sibson Robin
Published: December 2018

This series of papers is devoted to the investigation of the extent to which the accuracy of operation of multidimensional scaling methods can be put onto a quantitative footing. This first paper is concerned to provide a simple but clear and convincing development of procrustes analysis, the technique used for assessing the goodness-of-fit between two configurations. Although some of the results offered are new, it is largely a review paper designed to provide a basis for further work
Comparison of Two Sequential Tests for Two-Sided Alternatives
Authors: Borgan Ørnulf
Published: December 2018

Wald's method based on weight functions provides sequential tests for two-sided alternatives. In this paper, this method is compared with a test due to Sobel and Wald for three decision problems used for two-sided alternatives. The boundaries of acceptance and rejection of the tests are compared, and it is shown that for the usual choice of stopping bounds, the region of continued observation for Wald's test is included in that of Sobel and Wald's test. The difference between the region of rejection and acceptance of the two tests is also discussed.
The Construction of Resolution Iii Fractions from Generalized Cyclic Designs
Authors: Lewis S. M.
Published: December 2018

A simple method is described of constructing fractions of factorial experiments from generalized cyclic designs. A linear model approach is used to investigate the aliasing of effects.
Relative Curvature Measures of Nonlinearity
Authors: Bates Douglas M. , Watts Donald G.
Published: December 2018

Simple relative curvature measures of nonlinearity are developed to measure the extent of the nonlinearity in a model-experimental design-parameterization combination. We review the geometric aspects of linear and nonlinear least squares and, using the geometric concept of curvature, compute the maximum relative intrinsic curvature of the solution locus as well as the maximum relative parameter-effects curvature. The relative curvatures are independent of scale changes of the data and of the parameters so they can be used to compare different data sets as well as different parameterizations of the same data set. The methods are applied to 24 published nonlinear data sets, and in all cases it is found that the intrinsic curvature is less than the parameter-effects curvature. The nonlinearity measures of Beale (1960) and the bias expressions developed by Box (1971) are shown to be related to the curvature measures.
A Simple Method for Testing Goodness of Fit in the Presence of Nuisance Parameters
Authors: Braun Henry
Published: December 2018

This paper presents a method, based on the empirical distribution function, for testing goodness of fit (gf) under composite null hypotheses. After the unknown parameters are estimated from the entire data set, the procedure calls for the transformed sample to be randomly partitioned into a large number of groups, and a gf statistic calculated for each group. These statistics are used to construct a test which can attain, asymptotically, any desired level α, and which requires for its implementation only standard tables of critical values. The procedure is particularly recommended when an a priori grouping of the sample can be employed and, hence, heterogeneous alternatives are quite plausible. It is shown that, under these alternatives, the power of the procedure compares favourably with that of other methods.
Regression Models for Ordinal Data
Authors: McCullagh Peter
Published: December 2018

A general class of regression models for ordinal data is developed and discussed. These models utilize the ordinal nature of the data by describing various modes of stochastic ordering and this eliminates the need for assigning scores or otherwise assuming cardinality instead of ordinality. Two models in particular, the proportional odds and the proportional hazards models are likely to be most useful in practice because of the simplicity of their interpretation. These linear models are shown to be multivariate extensions of generalized linear models. Extensions to non-linear models are discussed and it is shown that even here the method of iteratively reweighted least squares converges to the maximum likelihood estimate, a property which greatly simplifies the necessary computation. Applications are discussed with the aid of examples.
A Generalized Bandit Problem
Authors: Nash Peter
Published: December 2018

A multi-armed bandit problem is investigated in which rewards obtained from pulls of any arm depend on the states of the other arms, as well as on the state of the arm pulled. A Dynamic Allocation Index is defined for this class of problems, and it is shown that this leads to optimal policies.
Order Determination for a Multivariate Autoregression
Authors: Quinn B. G.
Published: December 2018

The procedure proposed by Hannan and Quinn (1979) for determining the order of univariate autoregressions is extended to multivariate autoregressions. The procedure is shown to be strongly consistent, while other similar procedures may be shown to be strongly consistent by comparison.
Bayes Factors and Choice Criteria for Linear Models
Authors: Smith A. F. M. , Spiegelhalter D. J.
Published: December 2018

Global and local Bayes factors are defined and their respective roles examined as choice criteria among alternative linear models. The global Bayes factor is seen to function, in appropriate contexts, as a fully automatic Occam's razor and to be closely related to the Schwarz model choice criterion. The local Bayes factor is shown to have a close relationship with the Akaike Information Criterion.
Asymptotic Properties of Prediction Errors for the Multivariate Autoregressive Model Using Estimated Parameters
Authors: Reinsel Greg
Published: December 2018

In this paper we consider the asymptotic properties of the prediction errors for the multivariate autoregressive model when estimated coefficient parameters are used. Formulae for the asymptotic covariance matrix of the prediction errors for arbitrary lead times using estimated parameters are derived.
The Distribution of Higher-Order Interactions in Contingency Tables
Authors: O'Neill M. E.
Published: December 2018

Lancaster's definition of the absence of second-order interactions in contingency tables produces test statistics which have asymptotic ***x  2 distributions under complete independence, but not (in general) otherwise. This paper obtains the asymptotic distributions of all interaction components in the partitioning of Pearson's x  2, and indicates a test procedure for higher-order interactions when lower-order interactions are found to be significant.
The Effect of Survey Design on Regression Analysis
Authors: Nathan G. , Holt D.
Published: December 2018

Data arising from sample surveys are usually the result of complex survey design involving such techniques as stratification, multistage selection and the use of auxiliary information through unequal probability selection. These resulting data are often analysed using regression techniques without further regard to the sample design. This paper shows that, in general, ordinary least squares (OLS) regression will be biased in this situation even for large samples, although in the important class of most epsem designs asymptotic unbiasedness of least squares estimation is preserved. Alternative estimators are considered which yield unbiased estimates of the simple regression coefficient. Variances of all the estimators are derived and comparisons made. The usual OLS estimator of variance is also examined and found to be biased in general, even when used in conjunction with an epsem design for which the OLS estimator itself is unbiased.
A Note on Limit Theorems for Joint Distributions with Applications to Linear Signed Rank Statistics
Authors: Fligner Michael A.
Published: December 2018

When investigating the limiting joint distribution of a pair of random variables, it is possible to proceed by studying the associated marginal and conditional distributions. A useful extension of some previous work in this area of Sethuraman is presented by considering the important case when one is conditioning on a lattice-type random variable. The results are applied to obtain, in a simple manner, the limiting distribution of a linear-signed rank statistic.
Permutation Probabilities as Models for Horse Races
Authors: Henery R. J.
Published: December 2018

Some properties of models for the outcomes of races are described, these properties being consequences of a stochastic ordering of the permutations which define the outcomes of a race. Order statistics models which lead to stochastic ordering are also discussed—particular cases of these are the first-order model of Plackett (1975) and the normal model of Upton and Brook (1974). An approximation for the normal model is suggested.
Lagrange-Multiplier Tests of Multivariate Time-Series Models
Authors: Hosking J. R. M.
Published: December 2018

Generalizing Hosking (1980a), the Lagrange-multiplier test procedure is applied to hypotheses concerning multivariate autoregressive moving-average time-series models. The portmanteau and Quenouille goodness-of-fit tests for multivariate processes are derived in this manner and three other tests are obtained for multivariate moving-average models.
Distribution of the Residual Autocorrelations in Multivariate Arma Time Series Models
Authors: Li W. K. , McLeod A. I.
Published: December 2018

The large-sample distribution of the multivariate residual autocorrelations in the vector arma model is derived. This result is somewhat less complicated for the vector autoregressive model. A new multivariate portmanteau test for checking the adequacy of fitted vector arma models is developed. A simulation study shows that a simple modification of the portmanteau test improves its accuracy in small samples.
Multivariate Linear Relationships: Maximum Likelihood Estimation and Regression Bounds
Authors: Patefield W. M.
Published: December 2018

The well known result that, when the ratio of error variances is specified in normal theory bivariate linear relationships, the maximum likelihood estimator of the slope parameter takes the same form for both structural and functional relationships is extended to the multivariate case. Furthermore, it is well known that in the bivariate case the estimator of the slope is bounded by the two regression estimators. Analogous results are derived for the multivariate case.

The asymptotic variance–covariance matrix of the maximum likelihood estimators of the coefficients of a multivariate linear relationship is determined, and an estimator of this matrix is proposed. Large sample confidence regions for the parameters are formulated and the relationship between these and normal theory regression confidence regions is discussed.
Regression Diagnostics, Transformations and Constructed Variables
Authors: Atkinson A. C.
Published: December 2018

Two diagnostic displays for outlying and influential observations in multiple regression are reviewed. In some examples apparent outliers vanish after transformation of the data. Interpretation of the score statistic for transformations as regression on a constructed variable leads to partial residual plots for transformations and makes the two diagnostic displays available for detection of the influence of individual observations on the evidence for a transformation. Two forms of the partial residual plot are compared. A quick estimate of the transformation parameter is derived. This is shown to have poor properties and a superior sequence of estimates is found. The plots and estimates are studied for several examples using the power transformation for non-negative observations. The corresponding theory is developed for a “folded” transformation for proportions and percentages and an example given. The relationship with other constructed variables is explored, especially Tukey's one degeee of freedom for non-additivity. Pregibon's link test for generalized linear models is interpreted in terms of constructed variables and plots given for an example. This leads to the choice between two separate models for which a graphical Monte-Carlo procedure is suggested.
On Unequal Probability Sampling of Two Units Without Replacement
Authors: Agarwal S. K. , Kumar Pranesh , Dey A.
Published: December 2018

A general class of unequal probability sampling schemes for selecting two units without replacement is given. Many known sampling schemes are shown to be members of this class. It is shown that for any member of this class, the Yates-Grundy estimator of variance of the Horvitz-Thompson estimator is always positive.
The Thrown String
Authors: Kingman J. F. C.
Published: December 2018

The problem of modelling the configuration of a long flexible string thrown onto a table is considered, first statically, and then dynamically using ideas from classical statistical mechanics. The available experimental data are reviewed in the light of the theory.
Finding the Observed Information Matrix When Using the EM Algorithm
Authors: Louis Thomas A.
Published: December 2018

A procedure is derived for extracting the observed information matrix when the EM algorithm is used to find maximum likelihood estimates in incomplete data problems. The technique requires computation of a complete-data gradient vector or second derivative matrix, but not those associated with the incomplete data likelihood. In addition, a method useful in speeding up the convergence of the EM algorithm is developed. Two examples are presented.
Non-Parametric Bayesian Models for Samples from Finite Populations
Authors: Binder David A.
Published: December 2018

Using Ferguson's (1973) non-parametric priors in a Bayesian analysis of finite populations, we show that asymptotically, at least, the usual estimates and confidence intervals for the population mean in simple and stratified random samples can be justified in Bayesian terms. We then apply these models for estimating population percentiles, and a new procedure for interval estimates in stratified sampling is developed.
Exact D-Optimum Designs for Quadratic Regression
Authors: Gaffke N. , Krafft O.
Published: December 2018

It is shown that for quadratic regression on an interval a D-optimum design has its spectrum in the end- and mid-points of the interval with frequencies as equal as possible.
Orthogonal Series Distribution Function Estimation, with Applications
Authors: Hall Peter
Published: December 2018

Motivated by applications to the non-parametric estimation of mixture proportions, we develop techniques for multivariate orthogonal series estimation of distribution functions. We approximate the empiric distribution function, rather than the true distribution function. This enables us to use trigonometric series methods to estimate distribution functions with unbounded support.
Single Replicate Designs for Two-Factor Experiments
Authors: Lewis S. M. , Dean A. M. , Lewis P. H.
Published: December 2018

New single replicate designs are presented for experiments involving two quantitative factors. The designs have been selected from the individual replicates of resolvable generalized cyclic designs under the criterion of low loss of information on low degree components of the factorial effects.
Kernel-Based Density Estimates from Incomplete Data
Authors: Titterington D. M. , Mill G. M.
Published: December 2018

The problem of using non-parametric methods to estimate multivariate density functions from incomplete continuous data does not appear to have been considered before. Methods of producing kernel functions on incomplete observations are suggested involving averaging over the missing variables or substitution of them by simulated values. Consistency of the procedures in terms of integrated mean squared error is investigated and optimal choice of smoothing parameter is discussed. Application in terms of imputation for missing values is discussed.
A Kth Nearest Neighbour Clustering Procedure
Authors: Wong M. Anthony , Lane Tom
Published: December 2018

Due to the lack of development in the probabilistic and statistical aspects of clustering research, clustering procedures are often regarded as heuristics generating artificial clusters from a given set of sample data. In this paper, a clustering procedure that is useful for drawing statistical inference about the underlying population from a random sample is developed. It is based on the uniformly consistent kth nearest neighbour density estimate, and is applicable to both case-by-variable data matrices and case-by-case dissimilarity matrices. The proposed clustering procedure is shown to be asymptotically consistent for high-density clusters in several dimensions, and its small-sample behaviour is illustrated by an empirical example.
A Unified Way of Deriving Lmp Rank Tests from Censored Data
Authors: Basu A. P. , Ghosh J. K. , Sen P. K.
Published: December 2018

In this note a unified way of deriving locally most powerful rank tests for censored data is given. A lemma is proven. This lemma is then used to derive LMP rank tests.