bluuebunny/crossref_metadata_2025 huggingface dataset:
Methods of Estimation Involving Discounting
Authors: Gilchrist W. G.
Published: December 2018

In the estimation of the parameters in time series models arising in economic and business situations, it is often of value to introduce some method of discounting which produces updated estimators that give more weight to recent data than to that arising in the distant past. The simplest methods of introducing such discounting are to modify the traditional methods of estimation. Such modifications are discussed for the methods of moments, least squares and maximum likelihood. The final section discusses criteria upon which such techniques can be evaluated.
A Change-Over Design for Testing a Treatment Factor at Four Equally Spaced Levels
Authors: Berenblut I. I.
Published: December 2018

A new design for testing a quantitative factor at four equally spaced levels is presented. This design is suitable for quick analysis when the presence of first residual effects can be adequately accounted for by a single (linear) degree of freedom. The calculation for testing this assumption and the analysis in the general case are given.
On the Economic Choice of Experiment Sizes for Decision Regarding Certain Linear Combinations
Authors: Ericson W. A.
Published: December 2018

The author (1965) has presented an algorithm for choosing the sample sizes, ni (i = 1, …, k), to be selected from each of k independent normal data-generating processes having unknown means, μi, in order to minimize the posterior variance of an arbitrary given linear combination of these means subject to a budget constraint. The joint prior was taken to be a general k-dimensional normal distribution. In the present paper it is shown how, using this algorithm and for a wide class of loss functions, the optimal experimental design (choice of n1, …, nk) can be reduced to a univariate minimization problem. Several specific loss functions are considered as examples which provide generalization of decision models given by Raiffa and Schlaifer (1961), as well as a generalization of the earlier work of Grundy et al. (1956).
A Note on the Efficiency of Least-Squares Estimates
Authors: Cox D. R. , Hinkley D. V.
Published: December 2018

A linear model is considered in which errors are independent and identically distributed with zero mean. If the error distribution is specified, except possibly for unknown parameters, the asymptotic efficiency of least-squares estimates relative to maximum-likelihood estimates can be found. For regression parameters orthogonal to the general mean the asymptotic efficiency, which is independent of the design matrix, is calculated explicitly for an Edgeworth series, for a Pearson Type VII distribution and for a log gamma distribution.
The Occurrence of Replications in Optimal Designs of Experiments to Estimate Parameters in Non-Linear Models
Authors: Box M. J.
Published: December 2018

The occurrence of replications in optimal designs obtained by computer search is demonstrated for three non-linear estimation problems arising from the determination of chemical reaction rates. A theory is developed showing that some replicated designs are indeed optimal, whilst the rest, which have not been proved non-optimal, are certainly very good designs. Finally, recommendations are made as to the manner in which such experimental design problems might be attempted.
The Asymptotic Powers of Multivariate Tests with Grouped Data
Authors: McNeil D. R.
Published: December 2018

In many situations (see, for example, Bennett, 1962), random variables which may be assumed to have a multivariate normal distribution cannot be measured exactly, but are each given one of a discrete number (usually two) of scores. One is therefore concerned with the problem of making inferences about the parameters of a multivariate normal distribution when the data are grouped. In this paper we consider the problem of testing the hypothesis that the mean vector is zero, and we will compare the asymptotic powers of certain tests.
Models for the Response of a Mixture
Authors: Becker N. G.
Published: December 2018

Care needs to be exercised in the choice of model for a mixture system. The polynomial model, for example, cannot satisfactorily account for components which are inert or have additive effects and its coefficients lose their interpretative value when the variables are the proportions of components in the mixture. This paper proposes models constructed from functions homogeneous of degree one which overcome these difficulties, yet leave a wide choice of models.
A Unified Approach for Constructing a Useful Class of Non-Orthogonal Main Effect Plans in Kn Factorials
Authors: Raktoe B. L. , Federer W. T.
Published: December 2018

This paper presents a unified approach for constructing main effect plans in any kn factorial, where the k levels are equally spaced. The method consists of forming kn/2 pairs (when k is even) or (kn − 1)/2 pairs (when k is odd) of treatment combinations with the property that differences within pairs are clear of “even” effects and sums are clear of “odd” effects. Selecting kn/2 pairs (when k is even) or (kn − n + 2)/2 pairs (when k is odd) of treatment combinations in a prescribed manner lead to main effect plans such that “even” main effects are clear of “odd” main effects. General formulae of information matrices are presented for the 2n and 3n series.
Estimation of Parameters of a Finite Mixture of Distributions
Authors: Kabir A. B. M. Lutful
Published: December 2018

This paper is concerned with the estimation of the parameters of a finite mixture of distributions. The components of the mixture are assumed to belong to a subset of the exponential family of distributions. A procedure for estimation has been developed and the resulting estimators are shown to be consistent and asymptotically normally distributed. Finally, as applications, mixtures of two binomial and two exponential distributions have been considered.
Note on Discordant Observations
Authors: Hartigan J. A.
Published: December 2018

Given a set of observations x1, …, x  n, a Bayes measure of discordance of an observation x is defined to be the distance between the posterior distributions of a parameter, in the presence or absence of x. A measure of dissimilarity between two observations is also proposed. For large numbers of observations, these two measures may be approximated by simple functions of the log likelihood, thereby avoiding dependence on prior distributions. The theory is applied to data in which 13 judges ranked 20 mothers; each judge is supposed to give an independent observation on the mothers, with the analysis showing which judges are discordant from the rest and which judges are similar to each other.
On a Distribution Associated with Certain Stochastic Processes
Authors: Kemp Adrienne W. , Kemp C. D.
Published: December 2018

The distribution discussed is (a) the total number of games lost by the ruined gambler starting with a monetary units against an infinitely rich adversary in the classical ruin problem; (b) the total size of an epidemic in an infinite population starting with a cases and with constant probabilities of infection and recovery; (c) the number of customers served in a busy period (starting with a customers) of the queue M/M/1. These derivations, the relationships between them and various properties of the distribution are discussed.
An Extension of Quenouille's Test for the Compatibility of Correlation Structures in Time Series
Authors: Rosenhead Jonathan
Published: December 2018

Quenouille's test compares under stated assumptions the correlation structures of two lengths of time series to see if they are compatible. A test is here derived to compare any number of lengths of time series. As in Quenouille's test, the derivation depends on assumed or fitted values of autoregression coefficients; the test is shown to be insensitive to errors in these values.
Foundations for the Theory of Least Squares
Authors: Hill B. M.
Published: December 2018

The Bayesian theory of least squares is founded upon a weaker and more tangible form of prior knowledge than the conventional assumption of normality. The underlying assumption is a form of conditional uniformity on spheres for the “actual errors” in the experiment. This provides a unified theory appropriate for randomization models in the analysis of variance as well as for classical least-squares analysis.
A Note on an Allocation Problem
Authors: Scott Alastair
Published: December 2018

There is a fairly wide literature on choosing the size of an experiment when the cost of taking more observations is known. This paper shows that these results are useful in designing screening experiments where the size is fixed and the problem is to find the best way of dividing the limited resources between the treatments under test.
Regulation and Optimization
Authors: Dunsmore I. R.
Published: December 2018

A Bayesian statistical decision theory model is derived for some control problems in regression situations in which it is required to give a value of the concomitant variable x at which to perform an experiment in an effort to obtain a particular (although possibly unknown) value of the dependent variable y. In particular, problems of regulation and optimization are considered.
Experiments with Mixtures: Estimated Regression Function of the Multiple-Lattice Design
Authors: Lambrakis D. P.
Published: December 2018

Lambrakis (1968) introduced the multiple-lattice design for experiments with mixtures as a generalization of the simplex-lattice design introduced by Scheffé (1958), and derived the estimating regression function of the double-lattice design. In this paper an expression due to Savage (see Kiefer, 1961, footnote 6), which generates orthogonal polynomials on the points of a simplex-lattice, is used to derive the estimated regression function of any multiple-lattice design.
Compound Decisions and Empirical Bayes
Authors: Copas J. B.
Published: December 2018

The analysis of a sequence of decision problems of identical structure is considered, and the similarities and differences between the compound decision and empirical Bayes approaches explored. The paper includes a brief review of the literature and examines the criteria which have been used in comparing compound decision and empirical Bayes procedures with conventional methods. The performance of one of these procedures is examined in detail, and the previously accepted view that the compound decision approach is appropriate for situations outside those covered by empirical Bayes is questioned.
On Partitioning χ2 and Detecting Partial Association in Three-Way Contingency Tables
Authors: Goodman Leo A.
Published: December 2018

This paper presents a method of partitioning a χ2 statistic for the I × J × K contingency table (viz. the χ2 statistic that is based upon the likelihood-ratio criterion for testing the null hypothesis that the three variables pertaining to the three-way table are mutually independent) into additive components that can be used to test (1) the null hypothesis of zero three-factor interaction, (2) the null hypothesis that the partial association between two of the variables in the three-way table is zero, and (3) some null hypotheses concerning the two-way marginal distributions. This paper also discusses the estimation of the degree of partial association between two of the variables in the I × J × K table both in the case where the observed row and column marginals in each of the K different I × J tables (which form the I × J × K table) are considered given, and also in the case where these marginals are not fixed. Both the conditional maximum-likelihood estimator (given the observed row and column marginals in the K different I × J tables) and the unconditional maximum-likelihood estimator are presented, and for the special case where I = J = 2 various approximations to these estimators are compared.
Multi-Stage Interval Estimations of the Largest Mean of k Normal Populations
Authors: Tong Yung Liang
Published: December 2018

We have k normal populations (k ⩾ 1) with means θ1, θ2, …, θk and a common unknown variance σ2. A fixed-width confidence interval for θ*=max⁡1≤i≤k θi is desired so that the coverage probability is at least γ (preassigned) for every θ = (θ1, θ2, …, θk) and σ2 > 0. A class of multistage procedures is considered for the solution of this problem. It is shown that at least for k ≤ 2, the efficiencies of those procedures are always less than one.
A Simple Problem with Strikingly Different Frequentist and Bayesian Solutions
Authors: Chambers M. L.
Published: December 2018

Suppose that equal-sized samples are drawn from two normal populations which have unknown means but known and equal standard deviations. Then the paper considers the problem of making probabilistic statements about whether both population means exceed some lower bound. It seems natural to the author to expect procedures for computing this bound to have the property that the bound increases monotonically with the sample means observed. The paper, however, shows that whilst this is true of Bayesian procedures, it is, surprisingly, in general untrue of frequentist procedures.
On a Regenerative Phenomenon Occurring in a Storage Model
Authors: Prabhu N. U. , Rubinovitch Michael
Published: December 2018

Kingman's (1964) theory of regenerative phenomena is used to explain the nature of the empty period in a storage model proposed by Gani and Prabhu (1963).
Error Analysis by Replaced Samples
Authors: Hartigan J. A.
Published: December 2018

For normal observations {Xi, j = 1, …, N}, certain weighted sums Yi, are used to draw inferences about the sum Y of a further N observations. The same family of weights {Wij} are applied in error analysis of general computations t on general samples x1, …, xN, with approximate validity in a wide range of situations.
Block Designs and Missing Data
Authors: Pearce S. C. , Jeffers J. N. R.
Published: December 2018

The fitting of missing plot values is considered for the general case of block designs. A method is given for their direct calculation using the design matrices. An iterative method is recommended and is compared with others already in use. Two expressions are given for the covariance matrix of adjusted treatment means when data are incomplete.
A Test of Equality of Block-Diagonal Covariance Matrices and its Role of Unification
Authors: John S.
Published: December 2018

Let Cb, b = 0, 1, …, B, be the class of all d-dimensional matrices (i.e. having d rows and d columns) with a block-diagonal structure, the blocks along the diagonal being the identity matrix of order d1b, and mjb repetitions of a symmetric dsb-dimensional positive-definite matrix σsb, s = 2, 3, …, Sb. One or more of the dsb's may be zero. The model specifies initially that observations Wb, b ⩾ 1, are independent Wishart matrices of nb degrees of freedom and expected value nb σb, σb ∈ Cb. The hypothesis H0 to be tested is that σb, b ⩾ 1, are all equal to a matrix σ0 in C0. It is assumed that C0⊂C1∩C2∩…∩CB. The likelihood-ratio test criterion is obtained. Several current multivariate test criteria are shown to be special cases, but they do not exhaust the cases that are useful. From the Mellin transform of the test criterion, a χ2 series, a χ2 approximation, a non-central χ2 series and a non-central χ2 approximation are derived for the null distribtion of the test criterion. The error in the non-central χ2 approximation is of the order of n−3b, while the error in the χ2 approximation is of the order of n−2b.
Some Specific Results in Sequential Design
Authors: Bohrer Robert
Published: December 2018

Chernoff–Whittle and Paulson type decision rules are defined for a selection problem of some practical interest. Non-asymptotic risks of these rules are derived where possible. These are compared with asymptotic approximations, derived more generally. Examples motivate general conjectures about relative gain with good sequential designs and about convergence to the asymptotic case as functions of experimental informativeness and experimental flexibility.
Sampling Moments of Moments Associated with Univariate Distributions
Authors: Shenton L. R. , Bowman K. O. , Sheehan D.
Published: December 2018

This paper describes a simple computerized approach to the problem of finding the sampling moments of moments. Several applications to moment problems associated with univariate distributions (binomial, logarithmic series, gamma in particular) are mentioned.
The Relative Position of the Two Regression Curves
Authors: Finucan H. M.
Published: December 2018

In a non-singular bivariate Gaussian distribution the regression curve of y on x has smaller slope (measured by dy/dx) than the regression curve of x on y. Calling this the standard position and defining in a fairly obvious way a “clear-cut reversal” of it, we show that, for a distribution whose mean exists, such a reversal is impossible overall if the curves cross only once, and impossible at the extremities if the curves interlace in a more complicated way. Examples are given of distributions in which a ‘reversed’ configuration is found in a central portion of the range, and of pathological distributions (having no mean) with a reversed configuration everywhere.

Some extensions to the trivariate case are indicated.
Symmetric-Simplex Block Designs for Mixtures
Authors: Saxena S. K. , Nigam A. K.
Published: December 2018

In the present paper generalized block designs for experiments with mixtures of n components, where some or all the n components have non-zero proportions, have been obtained. These designs allow for the exploration of the simplex in any region which may be of interest. The block designs of Nigam (1970) turn out to be particular cases of the designs suggested here. The procedure outlined in this paper also permits the experimenter to have any number of blocks.
Comparison of Probability Distributions
Authors: Lindsey J. K.
Published: December 2018

By considering observed responses as discrete measurements, a method of comparison and plausibility of fit of probability distributions is developed using the multinomial distribution as the basis and using the likelihood function for making inferences. The proper interval widths are determined by a graphical method. A number of numerical examples are provided.
F-Independence and Null Correlation of Continuous, Bounded-Sum, Positive Variables
Authors: Darroch J. N. , James I. R.
Published: December 2018

The concept of F-independence, previously discussed (Darroch, 1971) for integer-valued variables, is here developed for r continuous, positive variables whose sum is bounded. Part of the theory parallels the theory for integer-valued variables and differs only in some additional mathematical questions which are encountered. Several topics discussed in this paper are special to continuous variables. These include the relationship of neutrality, of the Dirichlet distribution and of the beta distribution to F-independence. These relationships are effected through the concept of proportion-invariance. The paper concludes with an account of some correlation theory for bounded-sum variables.
A Total Entropy Criterion for the Dual Problem of Model Discrimination and Parameter Estimation
Authors: Borth David M.
Published: December 2018

A total entropy criterion is developed for the sequential design of experiments. The criterion is applicable to the dual problem of model discrimination and parameter estimation. The total entropy measures both the uncertainty about which mathematical model is correct and the uncertainty about the parameter vector for each model. The criterion is shown to lead to the choice of experiment for which the outcome is most uncertain, relative to the uncertainty due to experimental error.
Comments on the Efficient Use of Supplementary Information in Standard Sampling Procedures by E. K. Foreman and K. R. W. Brewer
Authors: Scott Alastair , Smith T. M. F.
Published: December 2018

This note demonstrates that supplementary information can be utilized more efficiently than indicated by Foreman and Brewer (1971).
A Sequential Test for Certain Composite Hypotheses
Authors: Whitehead John
Published: December 2018

A sequential procedure is developed to distinguish between null and alternative hypotheses of the type H  0: θ ≤ θ0 and H  1: θ ≥ θ1 (θ0 ≤ θ1) for one-parameter exponential families in such a way that the risks of type I and type II errors are smaller than some predetermined bounds. This is accomplished using observations of a sequence of random variables whose distribution is known up to the parameter θ, and involves the theory of certain tests with power one. The average sample number for this procedure is compared with that for Wald's sequential probability ratio test applied to the same problem.
Coefficients in D-Optimal Experimental Design
Authors: Sibson Robin , Kenny Alan
Published: December 2018

We give a new proof of an inequality on design measures due to Attwood; the proof given here allows the investigation of when equality is attained. The result obtained is that, in a D-optimal design for a problem satisfying mild regularity conditions, either the number of distinct levels equals the number k of parameters and each level receives weight 1/k, or the number of levels is greater and each receives weight strictly less than 1/k. A complementary result proved as a lemma explores the structure of the set of designs defined by such conditions. The consequences of these results for the construction of design algorithms are discussed briefly.
Non-Parametric Confidence Intervals for a Mean Using Censored Data
Authors: Breth M.
Published: December 2018

General Kolmogorov-type non-parametric confidence contours for an unknown continuous distribution function are converted into non-parametric confidence intervals for the mean of the distribution. It is shown how these estimation procedures may be modified to enable censored data to be handled.
On Alias Matrices and Generalized Defining Relationships of Equi-Information Factorial Arrangements
Authors: Raktoe B. L.
Published: December 2018

In many practical situations a fraction of the full replicate of a k  1 x k  2 x ... x kt factorial is used by an experimenter to estimate a subset of effects under the assumption that the rest of the effects are negligible. to get a deeper insight in what precisely is being estimated if the assumption of negligibility of the suppressed effects is not valid it is imperative that the experimenter has a complete knowledge of the aliasing structure. This paper shows how to compute the aliasing structure and hence the generalized defining relationship of any design in a particular class of equi-information designs, the class being generated by the action of the group of level permutations on a given design. It is shown that the calculations in the case of the 2' factorial are extremely simple and a practical example is given to illustrate the results.
Asymptotic Properties of the Wiener–Kolmogorov Predictor. Ii
Authors: Bhansali R. J.
Published: December 2018

The asymptotic properties of the autoregressive estimates a  (T)(u) obtained using the fles method (Bhansali, 1974) suggest that, in practice, a decision on truncating the a  (T)(u) at u = p, say, will need to be taken. This truncated sequence of the auto-regressive estimates is used to form alternative estimates of the linear predictor, the autoregressive spectral density function and the innovation variance. Some of the asymptotic properties of these estimates are derived for the case when the observed time series is a sample of an infinite, as well as a finite, order autoregressive process. A comparison of these estimates with the corresponding estimates obtained using the time domain autoregressive coefficients is carried out.
On the Nonparametric Estimation of Regression Functions
Authors: Benedetti Jacqueline K.
Published: December 2018

We consider a nonparametric technique proposed by Priestley and Chao (1972) for estimating an unknown regression function. Conditions for strong convergence and asymptotic normality are discussed. Special consideration is given to the optimal choice of a weighting function.
Recursive Estimation of the General Linear Model with Dependent Errors
Authors: McGilchrist C. A. , Sandland R. L.
Published: December 2018

The techniques for recursive estimation of the general linear model are extended to the case of dependent errors with known second-order properties. Recurrence relations for computation and a sequence of independent recursive residuals, useful for testing the constancy of the regression relation over time, are derived and studied. Computational savings are available when the error process is stationary.
Robust Estimation of Power Spectra
Authors: Kleiner B. , Martin R. D. , Thomson D. J.
Published: December 2018

Outliers in time series can play havoc with conventional power spectrum density estimation techniques when the spectrum features of interest consist of one or more peaks of relatively low amplitude. Furthermore, the outliers need not be large relative to the scale of the process in order to have considerable impact on the estimate; they only need to be large relative to the scale of the innovations process. Following a tutorial review of the central notions of robustness and the principles guiding conventional spectrum estimation, methods are presented for computing robust spectral densities. These are quite insensitive to such outliers and produce reliable estimates reflecting the structure of the core portion of the data. Two distinct methods are described. Both are based on robust prewhitening and on weighting the resulting residuals. They differ only in the way prewhitening is done: the first method computes the prewhitened values iteratively through a filtering operation; the second fits a robust autoregressive representation to the data by iterated weighted least squares. Each approach can be regarded as the solution to an intuitively appealing minimization problem, where the solution is exact in one case, approximate in the other. Both procedures have proved to be successful over a wide variety of data sets as will be demonstrated with real and simulated data.
A Remark on Systematic Latin Squares
Authors: Cox D. R.
Published: December 2018

Latin squares optimal for polynomial variation are noted.
An Influence Curve for Two-Sample Rank Tests
Authors: Eplett W. J. R.
Published: December 2018

An influence curve for two-sample linear rank tests is introduced. This influence curve can be viewed as a derivative of the limiting power and reflects the stability of the power to deviations from the assumed hypothesis testing model. It can also be interpreted in terms of the limit in probability of the test statistic and then it indicates the effect that small changes in the observations have upon the test statistic. The stability properties of several tests are examined using the influence curve. In view of the findings, two alternatives to the Normal scores test are studied. The influence curve appears to be useful both for detecting instability and for suggesting new, more robust, tests.
Exchange and Interchange Procedures to Search for Optimal Designs
Authors: Jones Byron , Eccleston John A.
Published: December 2018

An algorithm to search for optimal block designs is described. This algorithm consists of an exchange procedure and an interchange procedure. During the exchange procedure the “correct” treatment replications are determined and during the interchange procedure an optimal set of treatment concurrences is searched for. The optimality criterion is the sum of the weighted variances of a set of treatment contrasts of interest. Examples which illustrate how the algorithm can be used to obtain designs which have a variance structure that reflects the interest expressed in a set of treatment contrasts are given and the performance of the algorithm is discussed.
An Approximation to Certain Multivariate Normal Probabilities
Authors: Henery R. J.
Published: December 2018

In models of disease transmission, in which liability to disease is assumed to be normally distributed, the problem arises of calculating the probability that a given member of a group of individuals will contract the disease conditional on the other members exceeding some threshold. The special case in which the truncated variables are uncorrelated with each other occurs frequently and we consider here an approximation, based on a Gram-Charlier expansion, for the probabilities which arise in this case. In two dimensions the method is a serious rival to the tetrachoric series expansion only for large positive truncation levels. In higher dimensions other methods are either approximate or difficult to use.
The Estimability Structure of Linear Models and Submodels
Authors: Stewart I. , Wynn H. P.
Published: December 2018

When a group of parameters in a linear model is estimable it remains estimable under a smaller model “containing” those parameters. However, there are more subtle relationships between the estimability of models and submodels. Writing A ~ B if model A is estimable under B, the full properties of “ ~ ” are discussed. It is shown that they can all be derived by identifying a special sublattice of submodels namely those models which are fully inestimable in that every factor has at least one inestimable contrast. The use of the ideas in writing efficient programmes to discover the estimability structure is discussed and a complete list of all possible structures for three factor models given with examples.
The Fisher-Bingham Distribution on the Sphere
Authors: Kent John T.
Published: December 2018

The Fisher distribution is the analogue on the sphere of the isotropic bivariate normal distribution in the plane. The purpose of this paper is to propose and analyse a spherical analogue of the general bivariate normal distribution. Estimation, hypothesis testing and confidence regions are also discussed.
Non-Parametric Estimation for Non-Decreasing Lévy Processes
Authors: Basawa I. V. , Brockwell P. J.
Published: December 2018

Three non-parametric estimates of the measure corresponding to a non-decreasing Lévy process are discussed. Consistency and the limit distributions of the estimates are studied. The empirical process corresponding to one of the estimates is briefly discussed with an anticipation of an application to goodness-of-fit tests.
A Model for Association in Bivariate Survival Data
Authors: Oakes David
Published: December 2018

A reparameterization of a model introduced by D. G. Clayton for association in bivariate life-tables is discussed. Inference for the parameter governing the association is considered when the marginal distributions are specified up to Lehmann alternatives. The information matrix is derived explicitly and it is shown that the parameterization is moderately successful in introducing orthogonality between the association parameter and the two scale parameters. The likelihood proposed by Clayton for the case that the marginal distributions are completely unknown is criticized. An alternative nonparametric estimator based on Kendall's coefficient of concordance is proposed and its asymptotic variance evaluated.
A Note on Permutation Probabilities
Authors: Dansie B. R.
Published: December 2018

A generalization of the equivalence noted by Henery (1981) of a system of exponential order statistics to a model of Plackett (1975) is given.
Rounding Error in Regression: The Appropriateness of Sheppard's Corrections
Authors: Dempster Arthur P. , Rubin Donald B.
Published: December 2018

We consider three simple approaches to rounding error in least squares regression. The first treats the rounded data as if they were unrounded, the second adds an adjustment to the diagonal of the covariance matrix of the variables and the third subtracts an adjustment from the diagonal. The third, Sheppard's corrections, can be motivated as maximum likelihood with small rounding error and either (1) joint normal data or (2) normal residuals, “regular” independent variables, and large samples. Although an example and theory suggest that the third approach is usually preferable to the first two, a generally satisfactory attack on rounding error in regression requires the specification of the full distribution of variables, and convenient computational methods for this problem are not currently available.
Upper Bounds for Average Efficiency Factors of Two-Factor Interactions
Authors: Dean A. M. , Lewis S. M.
Published: December 2018

An upper bound is derived for the average efficiency factor of a two-factor interaction when an experiment is arranged as an incomplete block design. Attention is restricted to designs which have orthogonal factorial structure and which allow one or both main effects to be estimated with full efficiency. It is shown that several designs may be compared simultaneously using a graphical interpretation of the bound.
Outlier Resistant Distributions: Where Does the Probability Go?
Authors: Goldstein Michael
Published: December 2018

For a single observation x from a distribution with unknown location parameter θ, we investigate the relationship between the prior probability that θ lies in an interval I and the posterior probability that θ lies in I, given x, as x tends to infinity. In the case when the likelihood function is strictly unimodal, and thus outlier resistant, we give sufficient conditions to ensure that the limiting posterior probability is not greater than the prior probability, by exploiting dispersion properties of the distribution.
A Note on “Chi-Squared Tests with Survey Data”
Authors: Gross W. F.
Published: December 2018

In their paper “Chi-squared tests with survey data”, Holt et al. give a correction factor for the chi-squared test of independence from survey data. We obtain an expression for this factor in terms of the cell and marginal proportions and the asymptotic variances of their estimates.
On the Optimality and Usage of Reinforced Block Designs for Comparing Test Treatments with a Standard Treatment
Authors: Jacroux Mike
Published: December 2018

In this paper we consider the problem of determining optimal reinforced block designs for estimating the differences in effects between a set of test treatments of interest and some standard treatment. The optimality criterion considered is the average variance with which elementary treatment differences involving the standard treatment are estimated. A design is said to be reinforced with the standard treatment if each block of the design contains the standard treatment once, and we refer to such designs as SR-designs. Some sufficient conditions are derived for an SR-design to be optimal within the entire class of available SR-designs and some SR-designs which can be obtained by reinforcing each block of certain types of group divisible designs are shown to satisfy the conditions given. An efficiency criterion is also defined which can be used to compare arbitrary SR-designs with respect to how well they estimate treatment differences involving the standard treatment.
A Note on Restricted Maximum Likelihood Estimation with an Alternative Outlier Model
Authors: Thompson R.
Published: December 2018

For single outliers in normal theory fixed effects models a mean slippage model is commonly used. An alternative is to model the outlier as arising from an unknown observation with inflated variance. Maximum likelihood estimates for the position of the outlier under the two models need not agree. This paper considers maximizing a restricted part of the likelihood to estimate the variance parameters and characterizes these estimates in terms of standard least squares parameters. It is shown that the residual variance and outlier position are the same under both models.
Monotonicity of the Probability of Correct Selection or are Bigger Samples Better?
Authors: Bofinger Eve
Published: December 2018

The problem of selecting the normal population with largest mean when variances and sample sizes may be unequal is considered and conditions on the variances which ensure that the probability of correct selection increases when any sample size is increased are investigated. It is shown that the related probability of choosing a population with mean “close” to the largest does not have such a monotonic property while subset selection probabilities do. We conclude that “bigger samples are usually better”.
Comment on “Estimating Parameters in Continuous Univariate Distributions”
Authors: Titterington D. M.
Published: December 2018

It is remarked that the MPS estimation method of Cheng and Amin (1983) can be regarded as a maximum likelihood approach based on grouped data.
On the Estimation of Residual Variance and Order in Autoregressive Time Series
Authors: Paulsen Jostein , Tjøstheim Dag
Published: December 2018

We study the bias of Yule-Walker, least squares and Burg-type estimates of the residual variance of autoregressive processes. Both simulations and theory indicate that Yule-Walker estimates are inferior to least squares and Burg-type estimates. The effect on order determination is also studied, and we extend the results on overestimation of the AIC criterion to the general multivariate case. For strongly autocorrelated processes, Yule-Walker estimates of residual variance and order may be severely biased even for comparatively large sample sizes.
Re-Weighted Least Squares Estimation with Censored and Grouped Data: An Application of the EM Algorithm
Authors: Pettitt A. N.
Published: December 2018

We consider the problem of adapting weighted least squares estimation so that estimates can be found with censored or grouped data. The EM algorithm is implemented for the linear model where the errors are assumed to be generated by a scale mixture of normal distributions. In particular, explicit results are given for the case where the marginal distribution of the error is Student-t with an even number of degrees of freedom. Expressions for the observed information are found, relating grouped data results to those for right censored data. Some ad hoc rules for the identification of outliers are given. An example involving the lifetimes of motorettes is considered.
Perturbation Theory for Generalized Procrustes Analysis
Authors: Langron S. P. , Collins A. J.
Published: December 2018

Perturbation theory for Procrustes statistics is developed when both configurations are subject to independent normal errors. The total sum of squares is shown to partition into sums of squares for translation, rotation/reflection and scaling, plus a residual. These results are extended to generalized Procrustes analysis. This gives a theoretical justification to Gower's analysis of variance summary of the results from a generalized Procrustes analysis.
Multiparameter Linearization Theorems
Authors: Brown B. M.
Published: December 2018

A linearization theorem represents an estimating equation for an unknown parameter as approximately linear in the neighbourhood of the true parameter point, and can provide a convenient approach to asymptotic theory. A multiparameter version of a standard linearization theorem is proved, and applied to two examples: M-estimates in multiple regression, and the spherical median.
Diagnostics for Smoothing Splines
Authors: Eubank R. L.
Published: December 2018

Diagnostic measures appropriate for use with smoothing splines are derived and their properties are investigated. The proposed measures focus on detection of observations which substantially influence the fit and provide additional information over that obtained from examination of residuals alone. A numerical example illustrates the technique.
A Criterion for the Construction of Optimal Neighbour Designs
Authors: Williams E. R.
Published: December 2018

The model of Williams (1986) for the neighbour analysis of field trials is investigated in order to derive a criterion for the comparison of neighbour designs and hence the construction of optimal neighbour designs. A two-stage procedure is given whereby an efficient resolvable incomplete block design can be used as a starting point for the construction of an efficient neighbour design. A definition of balance for neighbour designs is presented. A comparison with the method of Wilkinson et al. (1983) is discussed.
A Model for High-Order Markov Chains
Authors: Raftery Adrian E.
Published: December 2018

A model for Markov chains of order higher than one is introduced which involves only one additional parameter for each extra lag. Asymptotic properties and the autocorrelation structure are investigated. Three examples are given in which the model appears to model data more successfully than both the usual high-order Markov chain and the alternative models of Jacobs and Lewis (1978), Pegram (1980) and Logan (1981).
On Asymptotic Normality of Limiting Density Functions with Bayesian Implications
Authors: Chen Chan-Fu
Published: December 2018

Precise statements of three basic conditions for the asymptotic normality of limiting density functions are given. In the Bayesian context, these conditions as a whole provide a greater generality and more operational flexibility for asymptotic posterior normality than those assumed by previous investigators.
Testing Collapsibility in Contingency Tables
Authors: Ducharme Gilles R. , Lepage Yves
Published: December 2018

In this article, the problem of testing for the collapsibility of a contingency table is studied. The notion of collapsibility is first discussed and three aspects of this property are distinguished. Tests for these three forms are presented, with attention focussed on 2 × 2 × K tables. Finally, it is shown that in a collapsible table, the marginal odds-ratio is more efficient than the usual estimators of the common odds-ratio.
Tests of Fit Based on Normalized Spacings
Authors: Lockhart R. A. , O'Reilly F. J. , Stephens M. A.
Published: December 2018

Normalized spacings provide useful tests of fit for many suitably regular continuous distributions; attractive features of the tests are that they can be used with unknown parameters and also with samples which are censored (Type 2) on the left and/or right. A transformation of the spacings leads, under the null hypothesis, to a set of z-values in (0, 1); however, these are not uniformly distributed except for spacings from the exponential or uniform distributions. Statistics based on the mean or the median of the z-values have already been suggested for tests for the Weibull (or equivalently the extreme-value) distribution; we now add the Anderson-Darling statistic. Asymptotic theory of the test statistics is given in general, and specialized to the normal, logistic and extreme-value distributions. Monte Carlo results show the asymptotic points can be used for relatively small samples. Also, a Monte Carlo study on power of the normal tests is given, which shows the Anderson-Darling statistic to be powerful against a wide range of alternatives; the mean and median can be non-consistent or even biased.
Penalized Likelihood Estimation of Hazard and Intensity Functions
Authors: Senthilselvan A.
Published: December 2018

A nonparametric estimate is proposed for the hazard function using penalized likelihood methods. The estimate turns out to be a hyperbolic spline function which is continuous with its first derivative discontinuous only at a finite number of points. It always assumes positive values, unlike Anderson and Senthilselvan’s (1980) quadratic spline estimate. This method is modified for the estimation of the intensity function of a nonstationary Poisson process. An example is given for comparison of the new estimator with the estimator of Anderson and Senthilselvan (1980).
Precision, Complexity and Bayesian Model Determination
Authors: Poskitt D. S.
Published: December 2018

In this paper a new criterion for model determination is proposed. An explicit decision theoretic approach to model selection is employed to derive a Bayes decision rule. The operational characteristics of the criterion are discussed and consistency is shown.
Stochastic Complexity
Authors: Rissanen Jorma
Published: December 2018

It is argued that all the useful information in observed data that can be extracted with a selected class of modeled distributions, will be obtained if we calculate the stochastic complexity, defined to be the shortest description length of the data. The same quantity also determines the greatest lower bound for prediction errors when the data are sequentially predicted. An abstract definition of stochastic complexity is given along with two fundamental theorems which justify the notion. Further, three explicit model selection criteria to approximate the stochastic complexity are described and the associated optimal models are interpreted to define asymptotically sufficient statistics for the data.
Design of Experiments for Studying Intergenotypic Competition
Authors: Azais J. M.
Published: December 2018

A method of construction and randomization of designs for experiments for studying intergenotypic competition is given. These designs are like row complete latin squares, but they include border plots, which are necessary under field conditions. These border plots permit intra-block randomization without destroying the neighbour-balance property. In several cases it is proved that the design is connected even under a general competition model.
Control of Stochastic Loss Networks, with Applications
Authors: Zachary Stan
Published: December 2018

We consider the application of the policy improvement routine of Markov decision theory to the control of a stochastic loss network which can process calls of differing types requiring differing resources. The objective is the minimisation of the cost of lost calls. In practice the derivation of exactly optimal policies is computationally intractable, and we identify circumstances in which nearly optimal policies may reasonably be computed. Examples are fully connected telephone networks and cellular radio networks.
Limit Distribution of a High Order Markov Chain
Authors: Adke S. R. , Deshmukh S. R.
Published: December 2018

The limit distribution of a Markov chain of order k > 1 is obtained under conditions weaker than those assumed by Raftery (1985). The results are shown to be valid also for arbitrary state space Markov sequences of order k > 1.
Gittins Procedures for Bandits with Delayed Responses
Authors: Eick Stephen G.
Published: December 2018

This paper introduces the multi-armed delayed response bandit with geometric discounting. The existence of dynamic allocation indices is shown when the discount factor is less than 1/2 or when the information bank size is zero. For the multi-armed delayed response bandit, the arm indicated by the dynamic allocation procedure or Gittins procedure is optimal when all information bank sizes are zero. A computational method for calculating indices is presented. The idea is to approximate the optimal strategy using a class of strategies whose worths are easy to calculate.
Nonparametric Estimation of the Probability of a Long Delay in the M/G/1 Queue
Authors: Gaver D. P. , Jacobs P. A.
Published: December 2018

A Poisson stream of customers with known arrival rate λ approaches a single server having independent identically distributed service times with unknown distribution. A nonparametric estimator of the probability of a long customer delay is obtained from an asymptotic renewal theoretic result giving an exponential approximation to the tail of the virtual waiting time distribution for a stable M/G/1 queue. Asymptotic properties of the estimator are obtained. Results of a simulation study of the small sample size behaviour are given.
Detecting Periodic Components in a White Gaussian Time Series
Authors: Chiu Shean-Tsong
Published: December 2018

A family of tests for periodic components in a white Gaussian series is proposed. The test is based on a statistic which is proportional to the ratio of the maximum periodogram to a trimmed mean of the periodograms. The asymptotic distributions of the statistics are obtained. It is shown that the proposed test and Fisher's test have the same asymptotic powers against the alternative of a sinusoidal wave at a non-zero Fourier frequency. The simulation results suggest that the proposed test is more powerful than Fisher's test against the alternatives of multiple peaks. The problem of detecting multiple peaks is also discussed.