bluuebunny/crossref_metadata_2025 huggingface dataset:
Multivariate Stochastic Differential and Difference Equations with Periodic Coefficients
Authors: Siskind V.
Published: December 2018

The theory of multivariate stochastic processes specified by linear matrix differential or difference equations is extended to the situation in which the coefficient matrices contain periodic terms. Questions of stationarity and ergodicity are discussed and algorithms for computing the autocovariance and spectral matrices given in outline. The paper concludes with a few remarks on applications and estimation problems.
An Integral Equation Approach to Equipment Failure
Authors: Downton F.
Published: December 2018

An integro-difference equation relating the distributions of the times between successive failures of a piece of complex equipment is obtained. It is shown that in the limit this equation yields the exponential distribution (or its discrete analogue, the geometric distribution). The way in which this limit is approached is discussed and some data on the successive times to failure of bus engines are examined in the light of the results obtained.
Linear Compounds of Normal Linear Regression Coefficients
Authors: Kabe D. G.
Published: December 2018

In the literature the double linear compounds of the multivariate linear normal regression coefficients are tested by using Roy's maximum root test, see, for example, Khatri (1962) and Morrison (1967, p. 109, equation (13)). The test statistic is Student's t. The t statistic gives narrower confidence bounds than those given by Roy's maximum root statistic. Morrison uses Hotelling's T2 statistic for testing linear compounds of the mean vector. However, the exact test statistic is Student's t.
An Explicit-Limit Theorem for the Critical Galton–Watson Process with Immigration
Authors: Seneta E.
Published: December 2018

For a critical Galton–Watson branching process with immigration, {Xn}, in which the offspring distribution has finite variance and the immigration distribution finite mean, it is shown that the random variables {Xn/n} approach a gamma distribution in law as n → ∞.
On Asymptotic Relative Efficiencies of a Class of Rank Tests
Authors: Gastwirth Joseph L.
Published: December 2018

A partial ordering is introduced in a class of rank tests according to the weight or emphasis they place on the middle ranks versus the extreme ones. This notion of order is shown to be related to van Zwet's partial ordering of symmetric densities according to the probability in their tails. Bounds on Pitman's asymptotic relative efficiency (a.r.e.) of several tests to the Wilcoxon test are established. Finally, a sharp lower bound for the asymptotic power of the Ψ-test is obtained.
Convergent Sequences of Bounds for Some Parameters of a Simple Sequential Test
Authors: Kemp Kenneth W.
Published: December 2018

A method is given which yields upper and lower bounds for the operating characteristic and average sample number of a sequential test designed to test a simple hypothesis against a simple alternative. As the substitution procedure described progresses, the sequence of bounds generated converge to the probability or a.s.n. to which they relate.
An Alternative Derivation of a Result Due to Srivastava and Bancroft
Authors: Arnold Barry C.
Published: December 2018

An alternative simple proof is presented of the inadmissibility (m.s.e.) of the usual unbiased estimate of the mean in a bivariate normal with known diagonal covariance matrix when the mean vector is restricted.
Distribution of Residual Autocorrelations in the Regression Model with Autoregressive–Moving Average Errors
Authors: Pierce David A.
Published: December 2018

In examining the adequacy of a statistical model an analysis of the residuals is often carried out; and for models involving a time series structure the serial correlations of the residuals are natural statistics to consider. In this paper it is shown that asymptotically the first s sample autocorrelations of the residuals, calculated from a least-squares fit of a regression model in which the errors follow a stationary autoregressive–moving average time series, possess a multivariate normal distribution. For moderately sized s this is nearly a singular distribution, with the “singularity” concentrated heavily on the autocorrelation coefficients of lowest order or lag. This distribution was first obtained by Box and Pierce (1970) for time series with no deterministic component; that it is independent of the regression structure of the model is shown to follow from general specification results of Durbin (1970), utilizing the large sample distribution of the parameter estimates recently obtained by the author (1971). Application of these results to testing model adequacy is also considered.
Some Families of Selection Probabilities for Sampling with Probability Proportional to Size
Authors: Dodds D. J. , Fryer J. G.
Published: December 2018

Three families of simple exact selection probabilities for sampling two units from N with probability proportional to size (p.p.s.) are described. All three schemes include the well-known Brewer solution as a special case. Using the Horvitz–Thompson statistic, the efficiencies of the schemes are calculated for some well-known populations, and it is shown that considerable variations in efficiency may exist within any family. Finally, some criteria for choosing a particular member of a family in practice are examined.
Weakly Stationary Point Processes and Random Measures
Authors: Daley D. J.
Published: December 2018

A weakly stationary point process on the real line is a point process with counting measure N(A) defined on Borel subsets A of the real line such that the first two moments of N(A) are finite for bounded Borel sets and invariant under translation. The expected number of points falling in an interval of length u equals μu and serves to define the intensity parameter μ of the process. The variance of the number of points falling in an interval of length u defines the variance function V(u) which, with μ, characterizes weakly stationary point processes. V(.) is absolutely continuous with its left- and right-hand derivatives differing on at most a countable dense set, with V′(0 +)= α=lim⁡θ→∞πG([O,θ])/θ and G(.) a uniquely determined σ-finite spectral measure on [0, ∞) in terms of which V(u)=∫[0,∞)(sin 12θu/12θ)2 G(dθ). These general results are also true of any weakly stationary signed random measure.

To any operation on a point process there corresponds an operation on the spectral measure, and the analogues of superposition, independent random motion of the points, deletion and multiplication of points and of forming cluster processes are exhibited.

The paper attempts to expound a rigorous basis for the general theory preceding Bartlett's (1963) development of spectral analysis for point processes.
Asymptotic Properties of Bayesian Decision Rules for Two Terminal Decisions and Multiple Sampling. Ii
Authors: Hald A. , Keiding N.
Published: December 2018

The paper is a continuation of a paper by the same authors.

The model is based on a differentiable prior distribution, a linear loss function, an asymptotically normal sampling distribution and sampling costs proportional to the sample size. Asymptotic expressions are derived for sample sizes, acceptance and rejection criteria and the minimum regret for some k-stage sampling and decision procedures.

In the present paper three cases not considered in the previous paper are treated. Some comments on the relation to sequential sampling, the relation to models for clinical trials and efficiency problems are also given.
Regression Models and Life-Tables
Authors: Cox D. R.
Published: December 2018

The analysis of censored failure times is considered. It is assumed that on each individual are available values of one or more explanatory variables. The hazard function (age-specific failure rate) is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time. A conditional likelihood is obtained, leading to inferences about the unknown regression coefficients. Some generalizations are outlined.
A Test for Time-Dependence of Linear Open-Loop Systems
Authors: Rao T. Subba , Tong H.
Published: December 2018

We consider the problem of testing a given open-loop system for time-dependence. The test is based on just a single realization of the input and a single realization of the output. It makes use of the concept of the “evolutionary cross-spectra” of a non-stationary vector process and rests essentially on testing the “uniformity” of a set of vectors, whose components consist of the “evolutionary gain-spectra” and “evolutionary phase-spectra”.

Using a logarithmic transformation on the evolutionary gain-spectra, we show that the mechanics of the test are formally equivalent to a two-factor multivariate analysis of variance (MANOVA) procedure.

Numerical illustrations of the proposed test procedure are included.
Linear Ordered Rank Tests Which are Asymptotically Efficient for the Two-Sample Problem
Authors: Deshpandé Jayant V.
Published: December 2018

In this paper the nonparametric two-sample problem is considered and tests are proposed for the hypothesis of identity of the two distribution functions. These tests are based on statistics which are linear combinations of ranks of the order statistics of one of the samples, in the two-sample rank order. It is seen that many existing statistics, proposed mainly on heuristic grounds, are members of this class. The asymptotic distribution of such statistics is discussed and those statistics which lead to tests with maximum ARE in this class are derived. It is seen that their ARE with respect to the “optimal” parametric tests is 1. In the appendix, we give the coefficients to be used for forming linear combinations which lead to the maximum ARE in certain specific cases.

The work reported here was carried out when the author was a Visiting Fellow at the University of Sheffield.
Minimum Risk Estimators with Applications
Authors: Swartz G. B. , Murthy V. K.
Published: December 2018

Methods for obtaining minimum risk estimators with various loss functions and restrictions on the class of estimators are presented. The results are applied to estimators using order statistics and to estimation by graphical procedures. Minimum risk estimators of the Weibull reciprocal-shape parameter that use two order statistics are given.
Non-Orthogonal Main-Effect Designs for Asymmetrical Factorial Experiments
Authors: Margolin Barry H.
Published: December 2018

Two techniques are presented for constructing efficient non-orthogonal main-effect designs. The first technique is an extended method of collapsing that permits non-orthogonality in the derived design. The second technique is a modification of the fold-over method of design construction to produce new main-effect plans. The designs in this paper supplement the major work on orthogonal asymmetrical main-effect plans by Addelman and Kempthorne. The new designs require fewer observations than the corresponding Addelman–Kempthorne plans; this advantage is gained at a sacrifice of orthogonality. The new plans are potentially useful in exploratory studies involving a few multi-level factors plus many two-level factors. Being more nearly saturated than the Addelman–Kempthorne plans, the new designs usually require graphical analysis or testing via a valid external measure of variability. The relationship of some of the new designs to Latin squares and Graeco-Latin squares is also discussed briefly.
The Number of Variables When Searching for an Optimum
Authors: Curnow R. N.
Published: December 2018

Reservations are expressed about the automatic inclusion of a large number of factors in factorial experiments. Assuming a quadratic response surface and using the expected yield at the estimated optimum as a criterion, approximate conditions on the relevant parameters are obtained under which one-variable experimentation is preferred to two-variable experimentation, and no experimentation is preferable to both. The probabilities of mistaking the sign of the curvature of a cross-sectional curve and of the whole surface are also compared. The choice of the best single variable for experimentation is discussed. Practical limitations of the approach are noted.
A Stochastic Model for Repeated Surveys
Authors: Blight B. J. N. , Scott A. J.
Published: December 2018

Patterson (1950) has derived estimates for the mean of a time-dependent population from a sequence of samples with partial replacement of units. These results are extended to situations in which the population mean follows a linear Markov process.
A General Bayesian Linear Model
Authors: Smith A. F. M.
Published: December 2018

Lindley and Smith (1972), using a hierarchical form of prior structure, derived Bayesian alternatives to least squares estimates for the linear model. This paper examines the Bayesian model in more detail and notes some general properties of the resulting estimators. Comparison is made with a least squares approach to models of the second kind, and an example is given.
Some General Points in the Theory of Optimal Experimental Design
Authors: Whittle P.
Published: December 2018

This paper is an expansion of comments made by the author in the discussion following the reading of papers by Laycock (1972) and Wynn (1972). Section 2 presents a direct proof and interpretation of the equivalence theorem for the general case of a concave criterion function. This has implications for iterative algorithms of the type suggested by Wynn, as indicated in Section 3. Section 4 is concerned with a characterization of D,-optimality as a criterion which preserves ranking of designs under certain transformations of the problem. Section 5 discusses quasi-linear representations of the criterion for c-optimality.
On Tukey's Confidence Intervals for the Contrasts in the Means of the Intraclass Correlation Model
Authors: Bhargava R. P. , Srivastava M. S.
Published: December 2018

In the intraclass correlation model when the variance σ  2 and the intraclass correlation coefficient 'p are both unknown, Tukey's confidence intervals for the contrasts in the means are obtained for the fixed sample and the sequential case. In the latter case confidence intervals of fixed width are obtained and the cost of not knowing σ2 and ‘p has been studied. From these results parallel results for Scheffé's confidence intervals can easily be obtained. A recommendation for the choice between the two is also made.
Evolutionary Spectra in Random Vibrations
Authors: Hammond J. K.
Published: December 2018

The evolutionary spectral representation for a class of non-stationary random processes is applied to vibrating systems. Input-response relations are developed for single degree of freedom and multimode systems involving time-varying spectra for both univariate and bivariate processes. Jet noise is modelled as an example of the univariate case and the response of an oscillator to this excitation is considered.
Bayesian Estimation of Box–Jenkins Transfer Function-Noise Models
Authors: Newbold Paul
Published: December 2018

The problem of estimating the coefficients of the transfer function-noise models proposed by Box and Jenkins is considered. A Bayesian approach is taken and the exact posterior density function of the coefficients is derived. Because of the complexity of this function, inference about individual parameters is virtually impossible except in the most simple cases.

Accordingly, it is desirable to consider certain approximations. In particular, it is shown that inference based on Student's t distribution provides an approximate solution to the estimation problem.
The Analysis of Covariance under a Randomization Model
Authors: Robinson J.
Published: December 2018

The randomization model for the analysis of covariance in a randomized block design is considered. Asymptotic properties of tests and estimates are obtained and it is shown that if the number of replicates is large, then the usual results concerning tests are approximately true. The large sample power of the tests are obtained and it is shown that using the concomitant variate increases the power.
Inference from Complex Samples
Authors: Kish Leslie , Frankel Martin Richard
Published: December 2018

The design of complex samples induces correlations between element values. In stratification negative correlation reduces the variance; but that gain is less for subclass means, and even less for their differences and for complex statistics. Clustering induces larger and positive correlations between element values. The resulting increase in variance is measured by the ratio deff, and is often severe. This is reduced but persists for subclass means, their differences, and for analytical statistics. Three methods for computing variances are compared in a large empirical study. The results are encouraging and useful.
On the Method of Maximum-Likelihood for Dependent Observations
Authors: Bhat B. R.
Published: December 2018

Under certain conditions on conditional probability densities, in the case of dependent observations, the maximum-likelihood estimate is proved to be consistent, asymptotically efficient and asymptotically normal.
Explicit Form of the Distribution of the Behrens-Fisher D-Statistic
Authors: Rahman M. , Saleh A. K. Md. Ehsanes
Published: December 2018

The distribution of the Berhens-Fisher d-statistic is given explicit form for all combinations of (v  1  v  2) the degrees of freedom whether odd or even, or one odd and one even. The density function of the d-statistic is expressed in terms of an Appell function which lends itself to easy computation of tail-probabilities. Values of 97.5 and 95 per cent points of the distribution are tabulated for chosen values of (v  1, v  2).
The Mean and Variance of Watson's Distribution
Authors: Dent Warren T. , Broffitt James D.
Published: December 2018

Watson's distribution is the distribution of the ratio of two linear combinations of independent x22 variables (the denominator being positive). Such ratios arise, for example, in the consideration of, or in approximations to, certain serial correlation coefficients. The moments of these ratios are known only in specialized cases. We derive the first two moments for the more general Watson distribution.
Planning Experiments for Discriminating between Models
Authors: Atkinson A. C. , Cox D. R.
Published: December 2018

Results in the theory of optimum regression designs are used to develop experimental designs for discriminating between several regression models. An extension of the Kiefer–Wolfowitz general equivalence theorem is used to establish properties of the procedures. A sequential data-dependent strategy is described and its properties investigated for specific examples. In conclusion, some difficulties in the Bayesian analysis of discrimination experiments are considered.
On Bird Navigation, with Some Statistical Undertones
Authors: Matthews G. V. T.
Published: December 2018

Young migratory birds appear innately programmed to fly for given distances in given directions. The orientations of the circular distributions of their escape attempts from cages, or of vanishing points in free flight, have been subjected to refined statistical analysis recently.

Time-compensated sun- and moon-orientations have been demonstrated. Compass directions are also obtained from the alignment of the constellations. Non-astronomical methods of orientation are of secondary importance.

Adult birds have the ability to correct for geographical displacement, sometimes over thousands of kilometres. A form of bi-coordinate navigation is probably involved, but the bases of the navigational grid remain uncertain. Pigeons can detect and react to small changes in sun altitude in terms of latitudinal displacement. The sun's position along its arc in relation to time may give longitudinal information.

Homeward orientation is not of phenomenal accuracy, leaving open the possibility that random search for the known area plays a part.
Environmental Impacts of Highway Traffic as Two-Sided Filtered Markov Renewal Processes
Authors: Marcus Allan H.
Published: December 2018

Traffic noise, microscale air pollution and other adverse environmental impacts of highways can often be represented as the superposition of the impacts of the individual vehicles. A Markov renewal process is often a useful representation of vehicle positions in multi-type “follow-the-leader” traffic flow. The impact of the vehicles is thus a filtered Markov renewal process whose moments can be calculated recursively using the method of L. Takacs and W. K. Smith.
Random Discrete Distributions
Authors: Kingman J. F. C.
Published: December 2018

It is impossible to choose at random a probability distribution on a countably infinite set in a manner invariant under permutations of that set. However, approximations to such a choice can be made by considering exchangeable probability measures on the class of probability distributions over a finite set, and letting the size of that set increase without limit. Under suitable conditions the resulting probabilities, when arranged in descending order, have non-degenerate limiting distributions. These apparently arcane considerations lead to rather concrete conclusions in certain problems in applied probability.
Asymptotic Approximations for Distributions of Ratios of Linear Functions of Random Variables
Authors: Broffitt J. D. , Dent W. T.
Published: December 2018

Normal asymptotic approximations for the distribution of the ratio of linear functions of independent and identically distributed random variables are derived. These approximations are checked for accuracy by comparisons with exact known distributions in specific examples. In particular, the examples considered are the distribution of the circular serial correlation coefficient and Watson's distribution. The comparisons suggest that for sample sizes as small as 10 or 20 the asymptotic results are most satisfactory.
On the Efficiency of Fisher's Tea-Tasting Designs
Authors: Morton R.
Published: December 2018

The Pitman efficiency is calculated for Fisher's tea-tasting designs under the assumption that the alternative hypothesis involves the lady observing random variables with an arbitrary distribution and a shift parameter. The lady's knowledge of the design prior to randomization is taken into account.
Row-And-Column Designs with Two Groups of Treatments Having Different Replications
Authors: Freeman G. H.
Published: December 2018

Row-and-column designs are described for two groups of treatments with different replications. One group of treatments may have only one member, but generally both groups contain several treatments. For most of the designs both groups of treatments are either orthogonal or balanced with respect to each of rows and columns separately, and such designs are classified into 23 types. There are also some designs requiring both rows and columns for balance. Methods are given for constructing the designs, for each type and for general classes within the types, where these exist. Some small designs are illustrated and the choice of designs for use in practice is discussed.
Techniques for Testing the Constancy of Regression Relationships Over Time
Authors: Brown R. L. , Durbin J. , Evans J. M.
Published: December 2018

Methods for studying the stability over time of regression relationships are considered. Recursive residuals, defined to be uncorrelated with zero means and constant variance, are introduced and tests based on the cusum and cusum of squares of recursive residuals are developed. Further techniques based on moving regressions, in which the regression model is fitted from a segment of data which is moved along the series, and on regression models whose coefficients are polynomials in time are studied. The Quandt log-likelihood ratio statistic is considered. Emphasis is placed on the use of graphical methods. The techniques proposed have been embodied in a comprehensive computer program, timvar. Use of the techniques is illustrated by applying them to three sets of data.
On the Concepts of Sufficiency and Ancillarity in the Presence of Nuisance Parameters
Authors: Dawid A. P.
Published: December 2018

Examples are presented to show how ambiguities may arise from attempts to define and apply analogues of sufficiency and ancillarity in the presence of nuisance parameters. It is necessary to tread with caution if one wishes to avoid inconsistencies or unexpected consequences inherent in principles which, on first acquaintance, appear unexceptionable.
Admissibility and Minimaxity of Some Maximum Likelihood Estimators When the Parameter Space is Restricted to Integers
Authors: Ghosh Malay
Published: December 2018

The present note studies the admissibility and minimaxity of maximum likelihood estimators in certain examples where the parameter space is restricted to integers.
Addition to the Nearest Whole Unit
Authors: Richardson J. W.
Published: December 2018

It is well known that, where a unit consists of a finite number of subunits, the distribution of the error committed by adding a sufficiently large number, n, of items rounded to the nearest whole unit, is approximately normal. This is true of both the probability function and the distribution function.

In this note, the differences from normality are obtained in the form of inequalities, of order n  −1; and, with certain minimal restrictions on the parameters, the upper bounds of departure from normality are shown to be strikingly small.
An Extension of the T-Method to General Unbalanced Models of Fixed Effects
Authors: Hochberg Yosef
Published: December 2018

1.

Introduction and Summary

Among the various procedures for simultaneous interval estimation, Tukey's T-method gives the shortest confidence intervals for pairwise contrasts when it is applicable. (Cf. Scheffé, 1959, Ch. 3; Miller, 1966, Ch. 2.) However, the T-method in its exact formulation (excluding one way anova, see Spjøtvoll and Stoline, 1973) is restricted to well-balanced designs.

In this paper we discuss a generalized T-type procedure for simultaneous interval estimation of all estimable parametric functions in general fixed effects, univariate linear models. The new method, which we subsequently abbreviate as the GT method, is an extension of a generalized T-procedure for the unbalanced one-way anova model given in Spjøtvoll and Stoline (1973).

In Section 2 we derive the new procedure in full-rank and in less than full-rank models. Some important features of the procedure are discussed in Section 3. In order to gain insight into the types of models where the new procedure is expected to produce satisfactory confidence intervals for pairwise contrasts, we illustrate, in Section 4, the application of the GT method in some unbalanced designs. In Section 5 we discuss a modified GT procedure which constitutes a combination of the GT1 in Hochberg (1974) and the GT methods. The proofs of some of the theorems are given in an Appendix for purposes of a convenient reading.
Factorization of Likelihood Functions for Full Exponential Families
Authors: Barndorff-Nielsen O.
Published: December 2018

Examples are presented of full exponential families for which the likelihood function factorizes into a product of functions in such a way that each of the factor functions depends on a subparameter only and these subparameters vary independently of each other. A number of results are then discussed, concerning the key case of just two factors, one being the marginal likelihood for a statistic u, the other the conditional likelihood given u. Any statistic u which effects such a factorization is called a cut. Various conditions are given for a statistic to be a cut, and a method for constructing regular exponential families with cuts is indicated.
An Invariance Property of the Poisson Distributions for a Hunted Immigrant Population
Authors: Abakuks Andris
Published: December 2018

A death process with immigration is considered, where some of the deaths are observable. It is shown that, starting with a Poisson prior distribution for the initial size of the population, the posterior distribution for the current size of the population is at all times also Poisson.
Histosplines with Knots Which are Order Statistics
Authors: Wahba Grace
Published: December 2018

The histospline considered here is a quadratic spline density estimate analogous to the Boneva–Kendall–Stefanov (BKS) histospline, with their equally spaced knots replaced by knots at every knth-order statistic. This estimate can be expected to be relatively more “flexible” where the density is greatest. We obtain “mean square error at a point” convergence rates. The rates obtained are uniform over classes of densities which have first (alternatively second or third) derivatives in a bounded set in ℒp. If kn is chosen optimally, then this estimate shares the same “near optimal” convergence rates of certain BKS estimates when the knot spacing is chosen optimally, kernel estimates when the scale factor is chosen optimally, and orthogonal series estimates when the length of the series is chosen optimally.
Resolvable Paired-Comparison Designs
Authors: Williams E. R.
Published: December 2018

A general method is given for the construction of resolvable paired-comparison designs for v treatments using the initial block of a cyclic incomplete block design for 12v treatments. The efficiency factor of the resulting design is shown to be a function of the efficiency factor of the cyclic design. Comparisons are made between the new designs and existing cyclic paired-comparison designs.
Generation of Factorial Designs
Authors: Patterson H. D.
Published: December 2018

This paper describes a method for generating asymmetrical factorial designs given specified relationships between the levels of treatment and plot factors. Comparisons are made with a generalized cyclic method (John and Dean, 1975) for single replicate designs.
Noniterative Least Squares Estimates, Standard Errors and F-Tests for Analyses of Variance with Missing Data
Authors: Rubin Donald B.
Published: December 2018

This article generalizes Rubin's method of least squares estimation of missing values in an analysis of variance. The general method produces not only least squares estimates of all parameters and the residual mean square, but also the correct least squares standard error and t-test for each contrast as well as the least squares sum of squares and F-test for each collection of contrasts. The method is non-iterative and requires only those subroutines designed to handle complete data plus a subroutine to find the inverse of an m x m symmetric matrix, where m is the number of missing values.
The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data
Authors: Turnbull Bruce W.
Published: December 2018

This paper is concerned with the non-parametric estimation of a distribution function F, when the data are incomplete due to grouping, censoring and/or truncation. Using the idea of self-consistency, a simple algorithm is constructed and shown to converge monotonically to yield a maximum likelihood estimate of F. An application to hypothesis testing is indicated.
Maximum Likelihood from Incomplete Data Via the EM Algorithm
Authors: Dempster A. P. , Laird N. M. , Rubin D. B.
Published: December 2018

A broadly applicable algorithm for computing maximum likelihood estimates from incomplete data is presented at various levels of generality. Theory showing the monotone behaviour of the likelihood and convergence of the algorithm is derived. Many examples are sketched, including missing value situations, applications to grouped, censored or truncated data, finite mixture models, variance component estimation, hyperparameter estimation, iteratively reweighted least squares and factor analysis.
Continuous Time Spectral Analysis of Intervals between Events in Stationary Point Processes
Authors: Charnock D. M.
Published: December 2018

The sequence of intervals between events in a stationary point process is considered as a process in continuous time. Expressions are given for a covariance function and the corresponding spectrum. Some examples are given to illustrate that the properties of the intervals as a process in continuous time are not the same as those of the intervals as a sequence. The methods developed are used to analyse a process consisting of times of R peaks in an electrocardiogram record. A remarkable similarity between this analysis and one based on the counting process is noted.
The von Mises–Fisher Matrix Distribution in Orientation Statistics
Authors: Khatri C. G. , Mardia K. V.
Published: December 2018

When n distinguishable directions in p dimensions are required to describe each orientation, Downs (1972) has extended the von Mises–Fisher distribution. We obtain the normalizing constant which leads to the investigation of various basic properties of the distribution. In particular, an explicit expression for the first population moment as well as the asymptotic distribution of the basic statistics are provided. The estimation problem, important testing problems, and exact sampling distributions are dealt with, and some techniques are applied to a set of vectorcardiogram data. An extension of the distribution is also proposed.
On the Moments of a Beta Function Transformation of a Poisson Variate
Authors: Veevers A. , Tweedie M. C. K.
Published: December 2018

A certain beta function of a Poisson variate has been shown to exhibit desirable variance-stabilizing properties comparing favourably with those of the more usual square-root transformations. This note reports an interesting relationship existing between the moments of these special forms of beta function of a Poisson variate and the standard normal ordinate and integral functions.
Spherical Matrix Distributions and a Multivariate Model
Authors: Dawid A. P.
Published: December 2018

We consider inference about the parameters of a multivariate linear model, in which the usual assumption of normality for the errors is replaced by a weaker assumption of spherical symmetry. Structural distributions and confidence regions are derived, and it is shown that inference about means is identical with that appropriate under normality, being based on a matrix generalization of “Studentization”. Some relevant distribution theory is developed, the approach throughout being “density-free”.
Time Savings in Censored Life Testing
Authors: Muenz Larry R. , Green Sylvan B.
Published: December 2018

It is often possible to achieve a saving of time in a life testing experiment by allowing right-censoring. This concept is important in clinical trials in which an early decision procedure is used. We describe the use of the ratio of order statistics as a measure of time savings, and give results on the distribution of such ratios. Numerical results are tabled and discussed for several underlying distributions of failure time (uniform, exponential, Pareto). We outline the application of this approach to the evaluation of early stopping procedures, with particular emphasis on analysing the time savings achieved by two-sample rank tests.
A Queueing Model for Traffic Flow
Authors: Gipps P. G.
Published: December 2018

A model for queue formation and disintegration is proposed for traffic on a road where flow is unimpeded. The model assumes that vehicles are travelling in random queues and a distribution for queue length is derived having as a single parameter the ratio of the rate of mergings between pairs of queues and the rate of overtaking from queues of more than one vehicle. The first moment and maximum likelihood estimator associated with the distribution are derived and compared in a simulation study. The distribution is then fitted to data collected in different sites, with varying results.
The Fast Fourier Transform Spectral Estimator
Authors: Jong Piet
Published: December 2018

This note shows how the Fast Fourier Transform (FFT) spectral estimator can be approximately written as an average of periodograms constructed at points 2πk/N removed, where k is an integer and N is the number of observations; it also shows the properties of this approximation.
On Goodness of Fit Tests Based on the Empirical Distribution Function for Uniform Spacings
Authors: Koziol James A.
Published: December 2018

In a review of tests based on spacings, Pyke (1965) constructs an empirical distribution function of uniform spacings, and suggests that goodness of fit problems could be investigated with Kolmogorov–Smirnov and Cramér–von Mises type statistics when applied to this empirical distribution function, were the appropriate exact or asymptotic critical values available. It is shown in this note that the limiting distributions of these statistics are indeed available, but that the powers of tests based on these statistics generally compare unfavourably with the powers of the corresponding tests applied to the usual empirical distribution function of order statistics. Components of a Cramér–von Mises statistic based on spacings are also briefly investigated.
Applications of Estimators of a Density and its Derivatives to Certain Statistical Problems
Authors: Singh R. S.
Published: December 2018

Several statistical problems are considered. By reducing them to problems of estimating functionals of a density, derivatives of a density, or both, a method of finding partial solutions is obtained.
Invariance Properties of Uncorrelated Residual Transformations
Authors: Godolphin E. J. , Tullio M.
Published: December 2018

The problem of obtaining an uncorrelated residual in the general linear model of full rank which is orthogonal to the design matrix has been discussed by Theil (1965), Brown et al. (1975) and others. In this paper it is shown that the problem is equivalent to stipulating an invariance requirement for the corresponding residual sum of squares, thus implying that alternative uncorrelated residuals are necessarily related by an orthogonal transformation. The set of uncorrelated residuals satisfying a given partitioning of the design matrix is derived. It is also shown that each member of this set enjoys an additional invariance property which enables it to be expressed as the same transformation of the vectors of observations, disturbances or least-squares residuals. Several implications of the results are discussed.
Curve Fitting and Optimal Design for Prediction
Authors: O'Hagan A.
Published: December 2018

The optimal design problem is tackled in the framework of a new model and new objectives. A regression model is proposed in which the regression function is permitted to take any form over the space X of independent variables. The design objective is based on fitting a simplified function for prediction. The approach is Bayesian throughout.

The new designs are more robust than conventional ones. They also avoid the need to limit artificially design points to a predetermined subset of X. New solutions are also offered for the problems of smoothing, curve fitting and the selection of regressor variables.
On the Half-Sample Method for Goodness-Of-Fit
Authors: Stephens M. A.
Published: December 2018

Two interesting processes, related to the empirical distribution function, have been pointed out by Rao (1972) and by Durbin (1973). The second, in particular, leads to the half-sample method, an elegant and simple technique for dealing with unknown parameters in goodness-of-fit testing, without the necessity of new tables of percentage points for each distribution tested. to those who have spent some effort in producing such tables, it is natural to wonder whether this effort has been in vain. In this paper we show that the answer is no.
Discrete Time Series Generated by Mixtures. I: Correlational and Runs Properties
Authors: Jacobs P. A. , Lewis P. A. W.
Published: December 2018

A broad but parametrically simple model for a stationary sequence of dependent discrete random variables is given and several submodels are discussed. The structure of the model is specified by the marginal distribution of the random variables and several other parameters. The sequence of random variables is formed by a probabilistic linear combination of i.i.d. discrete random variables and is in general not Markovian. Second-order joint moments and spectra are obtained for the model, as well as some properties for the lengths of runs. The special case of a process in which the variables take on only two values is considered; this binary process is useful as a model for the counting process in a discrete-time point process.
A Note on a Matrix Inversion in Multiple Exponential Smoothing
Authors: McKenzie E. , Peterson M. M.
Published: December 2018

In discussing the equivalence of certain polynomial predictors, Godolphin and Harrison (1975) note a matrix U employed in Multiple Exponential Smoothing whose inverse has not been obtained explicitly. The inverse of a related matrix p is obtained here and the relationship between the two matrices U and p reveals why no inverse has been obtained in general for U.
Confidence Intervals for the Quantiles of a Finite Population: Simple Random and Stratified Simple Random Sampling
Authors: Sedransk J. , Meyer J.
Published: December 2018

Given a finite population consisting of N units, it is desired to have a confidence interval for the (t/N)th quantile, Y  (t), of the population. Assuming simple random sampling, a confidence interval procedure is suggested and several expressions for the associated confidence coefficient are given. With stratified simple random sampling, for L = 2 strata, three confidence interval procedures are suggested. The associated confidence coefficients are derived under each of two (illustrative) assumptions about the nature of the stratification. (These assumptions refer to a probabilistic knowledge about the stratification.) The procedures are then compared using data from several finite populations.
On Birnbaum's Axiom of Mathematically Equivalent Experiments
Authors: Godambe V. P.
Published: December 2018

Birnbaum's (1962) axiom of mathematically equivalent experiments is discussed. An alternative, more compelling axiom, is put forward. This alternative axiom together with conditionality axiom does not imply likelihood principle.
Studies in the Robustness of Multidimensional Scaling: Perturbational Analysis of Classical Scaling
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 second paper investigates the response of the classical scaling method to small errors by making expansions in powers of the error term, and retaining the lowest non-cancelling power. Effects on individual eigenvalues and eigenvectors are considered, and shown to lead to useful auxiliary techniques for choice of dimensionality or for correction of bias. Procrustes statistics, as developed in the first paper of this series, are used to provide an overall picture of the error response; a perturbation theory for these is worked out, and is shown to lead to an approximate distribution theory. The details are worked out illustratively for a simple type of error structure.
Bayesian Inference for a Normal Dispersion Matrix and its Application to Stochastic Multiple Regression Analysis
Authors: Chen Chan-Fu
Published: December 2018

This paper discusses Bayesian inference procedures for a normal dispersion matrix. Structural information for the prior mean of the dispersion matrix is incorporated into the analysis through a Normal-Wishart prior distribution. Many of the resulting Bayes estimates are invariant, consistent and asymptotically efficient. Using this procedure, a coherent Bayesian argument for stochastic multiple regression analysis is developed, where dependent and independent variables are jointly random, without experimental control. It is shown that ordinary least squares, a general form of ridge regression, and factor analysis regression methods can be represented as special cases within this general framework. The preliminary report of a simulation study indicates that the Bayesian regression techniques demonstrate substantial improvements over least squares, even using frequentist criteria. Finally, the related problem of joint estimation of the normal population mean and dispersion matrix is briefly discussed in Section 5. The resulting Bayes estimate for the population mean is shown to be closely related to Stein estimates.
Complete Latin Squares and Related Experimental Designs
Authors: Freeman G. H.
Published: December 2018

A description is given of complete latin squares, which are squares with all ordered pairs of elements occurring next to each other once each in rows and columns, and of their uses as experimental designs. The existence of complete latin squares for all even numbers has been known for more than 30 years. Squares with various types of symmetry are considered here, all possible squares being described with side 4, 6, 8 and 10, together with some of side 12, 14, 16 and 18. These are put into genera whose members can all be obtained from each other by one or more of the processes of interchanging rows and columns, rearranging rows or reading the square in reverse. Where no complete latin squares exist, quasi-complete latin squares are considered in which all unordered pairs occur twice in rows and twice in columns. Such squares exist for all odd numbers, as well as even numbers greater than 4, and examples are given up to size 9.
Some Algebraic Properties of Ridge Coefficients
Authors: McDonald Gary C.
Published: December 2018

This note is concerned with one parameter ridge estimators in a multiple linear regression context as described by Hoerl and Kennard (Technometrics, 12 (1970), pp. 55-68). These estimators are rational functions of the ridge parameter, and some properties concerning sign changes and asymptotic behaviour are noted. Further properties of these estimators are derived assuming that the x'x matrix possesses intraclass (or uniform) correlation structure. Some of these characteristics, e.g. the order of the ridge coefficients and their rates-of-change, are unrelated to the degree of multicollinearity; while others, e.g. existence and location of sign changes, are explicitly related to multicollinearity.
Generalized Cyclic Designs with B < V as Duals of Cyclic Designs
Authors: John J. A.
Published: December 2018

It is shown that the best generalized cyclic design for b = n < v is obtained as the dual of a cyclic design.
On the Cliff-Ord Test for Spatial Correlation
Authors: Burridge P.
Published: December 2018

It is shown that the test for spatial correlation in regression disturbances can be derived by the application of Silvey's (1959) Lagrange Multiplier method.
Using Kernel Density Estimates to Investigate Multimodality
Authors: Silverman B. W.
Published: December 2018

A technique for using kernel density estimates to investigate the number of modes in a population is described and discussed. The amount of smoothing is chosen automatically in a natural way.
The G/G/1 Queueing System with a Particular Queue Discipline
Authors: Fakinos D.
Published: December 2018

This paper considers the G/G/1 queueing system under the assumption of a last-come, first served queue discipline where each customer begins service immediately upon his arrival. At the next arrival, the previous service is interrupted but no loss of service is involved. It is shown, among other things, that the equilibrium distribution of the number of customers in the system (when the system is considered exclusively at arrival epochs) is geometric.
On Random Sequences with Centred Spherical Symmetry
Authors: Smith A. F. M.
Published: December 2018

An infinite sequence of real random variables, any finite sample from which has a centred spherically symmetric joint distribution, is shown to have the property that, conditional on two further random variables, M and V, the variables are independent with the same normal distribution N(M, V).