bluuebunny/crossref_metadata_2025 huggingface dataset:
Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin
Authors: Cheng R. C. H. , Amin N. A. K.
Published: December 2018

A general method of estimating parameters in continuous univariate distributions is proposed. It is especially suited to cases where one of the parameters is an unknown shifted origin. This occurs, for example, in the three-parameter lognormal, gamma and Weibull models. For such distributions it is known that maximum likelihood (ML) estimation can break down because the likelihood is unbounded and this can lead to inconsistent estimators. Properties of the proposed method are described. In particular it is shown to give consistent estimators with asymptotic efficiency equal to ML estimators when these exist. Moreover it gives consistent, asymptotically efficient estimators in situations where ML fails. Examples are given including numerical ones showing the advantages of the method.
Regression and Ordered Categorical Variables
Authors: Anderson J. A.
Published: December 2018

A general approach to regression modelling for ordered categorical response variables, y, is given, which is equally applicable to ordered and unordered y. The regressor variables xT = (x  1  ,…,xp) may be continuous or categorical. The method is based on the logistic family which contains a hierarchy of regression models, ranging from ordered to unordered models. Ordered properties of the former, the stereotype model, are established. The choice between models is made empirically on the basis of model fit. This is particularly important for assessed, ordered categorical response variables, where it is not obvious a priori whether or not the ordering is relevant to the regression relationship. Model simplification is investigated in terms of whether or not the response categories are distinguishable with respect to x. The models are fitted iteratively using the method of maximum likelihood. Examples are given.
Symmetric Sequential Analysis: The Efficiencies of Sports Scoring Systems (With Particular Reference to Those of Tennis)
Authors: Miles Roger E.
Published: December 2018

Bernoulli-type models are adopted for the sequence of contested points between two players (or teams). In bipoints, there are two types of point – “A serving” and “B serving” – necessary for a realistic model of top (class) men's tennis. Unipoints is the special case of bipoints in which the probabilities that A (and hence B) wins each type of point are the same–a reasonable model for top men's squash rackets. Sports scoring systems based upon such play (“uniformats” and “biformats”) may be regarded as sequential statistical tests concerning the identity of the “better player”. Provided they satisfy certain fairness criteria, these tests are symmetric, with equal probabilities of type I and II error, and a unique efficiency p; then formally 1 – p represents the average fraction of play being virtually wasted through the use of a suboptimal scoring system.

Largely because in top men's tennis the proportions of service points won by both players are so high, the efficiency of traditional tennis scoring in such play is unduly low. Whilst the introduction of tiebreaker games has further reduced efficiency, it is shown how a simple entropy-increasing modification of the traditional scoring system serves to increase efficiency.
Monte Carlo Methods of Inference for Implicit Statistical Models
Authors: Diggle Peter J. , Gratton Richard J.
Published: December 2018

A prescribed statistical model is a parametric specification of the distribution of a random vector, whilst an implicit statistical model is one defined at a more fundamental level in terms of a generating stochastic mechanism. This paper develops methods of inference which can be used for implicit statistical models whose distribution theory is intractable. The kernel method of probability density estimation is advocated for estimating a log-likelihood from simulations of such a model. The development and testing of an algorithm for maximizing this estimated log-likelihood function is described. An illustrative example involving a stochastic model for quantal response assays is given. Possible applications of the maximization algorithm to ad hoc methods of parameter estimation are noted briefly, and illustrated by an example involving a model for the spatial pattern of displaced amacrine cells in the retina of a rabbit.
Recursive Parameter Estimation Using Incomplete Data
Authors: Titterington D. M.
Published: December 2018

Stochastic approximation procedures are considered for the estimation of parameters using incomplete data. One procedure is stated and illustrated which often leads to asymptotically efficient estimators. Others are developed which, although possibly not optimal in the above sense, will be very much easier to apply. This will be particularly advantageous when quick recursive estimates are required. Examples are given and a link is made between one of the sub-optimal methods and the EM algorithm.
A Test for Comparing Estimators Under the Generalized Mean Square Error Criterion
Authors: Price J. Michael
Published: December 2018

This paper presents an asymptotic test for comparing alternative estimators of regression coefficients in terms of the generalized mean square error criterion. The paper also reports the outcome of a Monte Carlo experiment which was conducted to determine the small sample properties of the test statistic. These results indicate that the asymptotic test may be applied successfully even when the sample size is small.
Outlier Models and Prior Distributions in Bayesian Linear Regression
Authors: West Mike
Published: December 2018

Bayesian inference in regression models is considered using heavy-tailed error distributions to accommodate outliers. The particular class of distributions that can be constructed as scale mixtures of normal distributions are examined and use is made of them as both error models and prior distributions in Bayesian linear modelling, including simple regression and more complex hierarchical models with structured priors depending on unknown hyperprior parameters.
On the Analysis of Heterogeneity Among Populations
Authors: Gilula Zvi
Published: December 2018

A set of populations, each of which is characterized by a distribution, is said to be “overall homogeneous” if the underlying distributions are identical. Assuming that the underlying distributions are normal or multinomial, this paper provides a theory of establishing test statistics for testing hypotheses of homogeneity within subsets of populations and of homogeneity between subsets of populations. The desired test statistics are derived by a decomposition of the test statistic for the hypothesis of overall homogeneity into a sum of mutually statistically independent components.
Changeover Designs and Factorial Structure
Authors: Fletcher D. J. , John J. A.
Published: December 2018

Existing changeover designs given in the literature are not necessarily suitable when the treatments come from a factorial experiment. The concept of factorial structure in changeover designs is defined and shown to be a desirable requirement for these types of designs. Different classes of designs are then examined in the light of this concept. In particular, it is shown that all generalized cyclic (GC/n) changeover designs have factorial structure, thereby providing a flexible class of useful designs.
Efficiency Bounds for Partially Balanced Change-Over Designs Based on M-Associate Class PBIB Designs
Authors: Raghavarao D. , Blaisdell E. A.
Published: December 2018

In an earlier paper the authors introduced PBCOD(m) designs, which are change-over designs that are partially balanced for the estimation of the elementary direct and residual effects contrasts, and whose constructions are based on m-associate class PBIB designs. In this paper we determine the maximum efficiencies for the estimated direct and residual elementary treatment contrasts of PBCOD(m), and the A-optimality (Kiefer, 1959) of these designs is explored.
A General Class of Distributions on the Simplex
Authors: Aitchison J.
Published: December 2018

The Dirichlet class of distributions on the simplex is overstructured for practical work because of its many strong independence properties. A persistent problem of distribution theory has thus been to extend this class to include members free of these independence properties. An alternative approach has been the recent introduction of the logistic-normal class which contains members with and without the independence properties. Unfortunately, this class is separate from the Dirichlet class and so cannot sustain discussion of some strong forms of independence. These difficulties are overcome in this paper by the introduction of a more general class which includes as special cases the Dirichlet and logistic-normal classes. The role of this new class in relation to the analysis of compositional data, in particular the investigation of independence hypotheses, is discussed and illustrated.
Modelling and Residual Analysis of Nonlinear Autoregressive Time Series in Exponential Variables
Authors: Lawrance A. J. , Lewis P. A. W.
Published: December 2018

An approach to modelling and residual analysis of nonlinear autoregressive time series in exponential variables is presented; the approach is illustrated by an analysis of a long series of wind velocity data which has first been detrended and then transformed into a stationary series with an exponential marginal distribution. The stationary series is modelled with a newly developed type of second order autoregressive process with random coefficients, called the NEAR(2) model; it has a second order autoregressive correlation structure but is nonlinear because its coefficients are random. The exponential distributional assumptions involved in this model highlight a very broad four parameter structure which combines five exponential random variables into a sixth exponential random variable; other applications of this structure are briefly considered. Dependency in the NEAR(2) process not accounted for by standard autocorrelations is explored by developing a residual analysis for time series having autoregressive correlation structure; this involves defining linear uncorrelated residuals which are dependent, and then assessing this higher order dependence by standard time series computations. The application of this residual analysis to the wind velocity data illustrates both the utility and difficulty of nonlinear time series modelling.
A Note on Efficiency-Consistent Designs
Authors: Lewis S. M. , Dean A. M.
Published: December 2018

Every equi-replicate connected design with orthogonal factorial structure is shown to be efficiency-consistent. This result implies that all such designs are faithful with regard to main effects.
On Palm's Model with a Large Number of Machines and Repairmen
Authors: Salia R. N. , Shashiashvili M. A.
Published: December 2018

It is established in this note that the stationary probabilities of Palm's model for a large number of machines and repairmen (and the relation between them) can be approximated by a probability density, which is the “pasting together” of two normal densities with different variances.

Further, this result is used to find o(n  ½)-optimal number of repairmen in the problem of minimization of the expected costs of production.

In the last part of the paper, in connection with the investigation of the asymptotics of the operative efficiency, some results by Naor (1957) are discussed.
An Exact Chi-Squared Decomposition Theorem for Inverse Gaussian Variates
Authors: Letac G. , Seshadri V. , Whitmore G. A.
Published: December 2018

It is proved that the scaled deviance of an inverse Gaussian sample of size n can be expressed as a sum of n – 1 independent chi-square variates. Applications to prediction intervals and squared recursive-residuals are illustrated.
On Auto-Repeat Facilities and Telephone Network Performance
Authors: Kelly F. P.
Published: December 2018

New developments in telecommunications technology are likely to lead to substantial increases in repeat-attempt rates and this may well adversely affect the performance of the telephone network. One aspect of this complex problem is considered in this paper, where we explore how the characteristics of a multi-server system are affected by the proportion of demands that are prepared to wait for service.
Assessment of Local Influence
Authors: Cook R. Dennis
Published: December 2018

Statistical models usually involve some degree of approximation and therefore are nearly always wrong. Because of this inexactness, an assessment of the influence of minor perturbations of the model is important. We discuss a method for carrying out such an assessment. The method is not restricted to linear regression models, and it seems to provide a relatively simple, unified approach for handling a variety of problems.
The Statistical Behaviour of Some Least Squares Estimators of the Centre and Radius of a Circle
Authors: Berman Mark , Culpin David
Published: December 2018

This paper examines the statistical behaviour of some least squares estimators of the centre and radius of a circle. Two error models are used. The asymptotic consistency of the estimators is investigated. Where asymptotic consistency is established, asymptotic covariance matrices are obtained. A small sample simulation study gives results showing the same pattern as those of the asymptotic theory.
A Note on Bayes Factors for Log-Linear Contingency Table Models with Vague Prior Information
Authors: Raftery A. E.
Published: December 2018

The approximate Bayes factor, B  01, for log-linear contingency table models proposed by Spiegelhalter and Smith (1982) is indeterminate if any of the cell frequencies is zero. It is noted that use of a standard Jeffreys prior overcomes this difficulty. It is pointed out that – 2 log B  01 is approximately equivalent to Schwarz's (1978) model selection criterion in large samples.
A Note on the Calculation of Bartlett Adjustments
Authors: Barndorff-Nielsen O. E. , Blaesild P.
Published: December 2018

We describe, for the numerical calculation of Bartlett adjustments, a method which may be of use when the cumulants of the log likelihood derivatives are easy to determine in one parametrization while the hypotheses to be tested are all linear in some other parametrization. The method relies on the cartesian tensorial nature of those cumulants and should be particularly convenient in connection with statistical packages with structure similar to GLIM.
Parameter Orthogonality and Approximate Conditional Inference
Authors: Cox D. R. , Reid N.
Published: December 2018

We consider inference for a scalar parameter Ψ in the presence of one or more nuisance parameters. The nuisance parameters are required to be orthogonal to the parameter of interest, and the construction and interpretation of orthogonalized parameters is discussed in some detail. For purposes of inference we propose a likelihood ratio statistic constructed from the conditional distribution of the observations, given maximum likelihood estimates for the nuisance parameters. We consider to what extent this is preferable to the profile likelihood ratio statistic in which the likelihood function is maximized over the nuisance parameters. There are close connections to the modified profile likelihood of Barndorff-Nielsen (1983). The normal transformation model of Box and Cox (1964) is discussed as an illustration.
Confidence and Conflict in Multivariate Calibration
Authors: Brown Philip J. , Sundberg Rolf
Published: December 2018

Multivariate calibration involves using an estimated relationship between a multivariate response Y and an explanatory vector X to predict unknown X in future from further observed responses. In controlled calibration with multivariate normal errors, the profile likelihood function for the unknown X (denoted ξ) displays a term which measures the mutual inconsistency of the given response vector (denoted Z) in predicting ξ. This inconsistency diagnostic fundamentally differentiates the behaviour of likelihood based and Bayes “confidence” regions from those of the unconditional sampling approach. In addition the diagnostic serves to pinpoint an inadequate response vector Z.
On the Bootstrap and Continuity Correction
Authors: Hall Peter
Published: December 2018

A continuity correction to the bootstrap is suggested, useful when the bootstrap is employed to approximate distributions of lattice-valued sample means. Versions of the correction are available quite generally, even in multivariate cases. Asymptotic theory reveals that the correction can reduce maximum rounding error by up to 50%. An example, where continuity correction is used in the construction of confidence intervals, illustrates the performance of the procedure with smaller samples.
Asymptotic Posterior Normality for Stochastic Processes Revisited
Authors: Sweeting T. J. , Adekola A. O.
Published: December 2018

The problem of demonstrating the limiting normality of a posterior distribution arising from some stochastic process is revisited. It is shown that certain processes of practical interest are not covered by conditions currently available in the literature. In this paper we present a fairly general set of conditions for asymptotic posterior normality which cover a wide class of problems. The theory is applied to a nonhomogeneous Poisson process and birth process.
Estimation and Inference by Compact Coding
Authors: Wallace C. S. , Freeman P. R.
Published: December 2018

The systematic variation within a set of data, as represented by a usual statistical model, may be used to encode the data in a more compact form than would be possible if they were considered to be purely random. The encoded form has two parts. The first states the inferred estimates of the unknown parameters in the model, the second states the data using an optimal code based on the data probability distribution implied by those parameter estimates. Choosing the model and the estimates that give the most compact coding leads to an interesting general inference procedure. In its strict form it has great generality and several nice properties but is computationally infeasible. An approximate form is developed and its relation to other methods is explored.
A Note on the Aitkin-Rubin Approach to Hypothesis Testing in Mixture Models
Authors: Quinn B. G. , McLachlan G. J. , Hjort N. L.
Published: December 2018

In testing for the number of components in a mixture, it is well known that regularity conditions do not hold for minus twice the log likelihood ratio to have its usual asymptotic null distribution of chi-squared. In an attempt to overcome some of the problems, it has been proposed in the literature that a prior distribution be placed on the mixing proportions. We show here that the usual regularity conditions still do not hold.
Tightness of Monotone Random Fields
Authors: Ritov Y.
Published: December 2018

For monotone random fields (e.g. gradients of convex functions) pointwise convergence implies uniform convergence on bounded subsets. This result was used in Brown (1985) but an incorrect proof was given. We give a new proof.
Symmetry Models and Hypotheses for Structured Data Layouts
Authors: Dawid A. P.
Published: December 2018

The structure of a data layout may be usefully described in terms of its symmetries, i.e. transformations which do not affect attitudes to the data. Interesting null hypotheses may be considered as imposing still further symmetries. These symmetry properties may be used to derive valid tests of such null hypotheses, and to construct appropriate statistical models for the data. The general ideas are developed and illustrated in the context of a replicated two-way classification.
Consistent Order Determination for Processes with Infinite Variance
Authors: Bhansali R. J.
Published: December 2018

Order determination for autoregressive processes when the residuals have a distribution belonging to the domain of attraction of an infinite variance stable law is examined. The use of the FPEα criterion of Bhansali and Downham (1977) is shown to provide a consistent estimator of the order with any α > 0. Order determination for moving average processes is considered; a procedure based on autoregressive estimates of the inverse correlations, see Bhansali (1983), is shown to provide a consistent estimator of the order. Consistency of the estimated autoregressive and moving average parameters of an arbitrary order is established. The question of consistent discrimination between autoregressive and moving average models is also considered. The finite sample behaviour of the discrimination procedure is investigated by means of a simulation study.
A Note on Extended Quasi-Likelihood
Authors: Davidian M. , Carroll R. J.
Published: December 2018

We study the method of extended quasi-likelihood estimation and inference of a variance function recently proposed by Nelder and Pregibon. The estimates are inconsistent in general, and the test levels can be biased, but in many cases such as the exponential family the inconsistency and bias will not be a major concern. Extended quasi-likelihood is compared with Carroll and Ruppert's pseudo-likelihood method, which gives consistent estimates and, when slightly modified, asymptotically unbiased tests. We quantify the notion of a problem in which the amount of statistical information is large in each unit, showing in this instance that the two estimates are closely related and may be asymptotically equivalent in many important cases. However, in some cases outside the exponential family, an asymptotic bias can persist.
Balanced Polycross Designs
Authors: Morgan J. P.
Published: December 2018

Methods for constructing non-directional balanced polycross designs are presented, the balance being over row, column, and diagonal neighbours. The idea of a terrace is generalised and used to construct a set of arrays which enjoys complete balance. The arrays are then combined to obtain smaller, balanced designs; explicit constructions are given for ν < 20 varieties. Many of the designs are in v or v + 1 replicates and have treatments balanced in rows and columns. As an additional benefit, some of the EBIBD designs of Kiefer and Wynn (1981) are produced.
Consistent Estimates for the Near(2) and Nlar(2) Time Series Models
Authors: Karlsen Hans , TjøStheim Dag
Published: December 2018

Consistent and asymptotically normal estimates for all four parameters of the NEAR(2) and NLAR(2) time series models are obtained. The estimates are given in explicit form and are easy to compute, but with simulation experiments the standard errors are large, especially for small values of the parameters, so very substantial sample sizes may be needed. Hence, from this point of view, the NEAR(2) and NLAR(2) models are of limited practical value. The simulations indicate, however, that reasonably good estimates can be obtained for sample sizes in the range used by Lawrance and Lewis in their investigation of a series of wind data.
Quantifying Expert Opinion in Linear Regression Problems
Authors: Garthwaite Paul H. , Dickey James M.
Published: December 2018

This paper describes a method for choosing a natural conjugate prior distribution for a normal linear sampling model. A person using the method to quantify his/her opinions performs specified elicitation tasks. The hyperparameters of the conjugate distribution are estimated from the elicited values. The method is designed to require elicitation tasks that people can perform competently and introduces a type of task not previously reported. A property of the method is that the assessed variance matrices are certain to be positive definite. The method is sufficiently simple to implement with an interactive computer program on a microcomputer.
Leave-K-Out Diagnostics for Time Series
Authors: Bruce Andrew G. , Martin R. Douglas
Published: December 2018

We propose diagnostics for autoregressive integrated moving average (ARIMA) model fitting for time series formed by deleting observations from the data and measuring the change in the estimates of the parameters. The use of leave-one-out diagnostics is a well-established tool in regression analysis. We demonstrate the efficacy of observation-deletion-based diagnostics for ARIMA models, addressing issues special to the time series setting. It is shown that the dependency aspect of time series data gives rise to a ‘smearing’ effect, which confounds the diagnostics for the coefficients. It is also shown that the diagnostics based on the innovations variance are much clearer and more sensitive than those for the coefficients. A ‘leave-k-out’ diagnostics approach is proposed to deal with patches of outliers, and problems caused by ‘masking’ are handled by use of iterative deletion. An overall strategy for ARIMA model fitting is given and applied to two data sets.
The Effect of Simulation Order on Level Accuracy and Power of Monte Carlo Tests
Authors: Hall Peter , Titterington D. M.
Published: December 2018

We discuss the effect of simulation order on level accuracy and power of Monte Carlo tests, in a very general setting. Both parametric problems, with or without nuisance parameters, and nonparametric problems are treated by a single unifying argument. It is shown that if the level of a Monte Carlo test is known only nominally, not precisely, then the level error of a Monte Carlo test is an order of magnitude less than that of the corresponding asymptotic test. This result is available whenever the test statistic is asymptotically pivotal, even if the number of simulations is held fixed as the sample size n increases. It implies that Monte Carlo methods are a real alternative to asymptotic methods. We also show that, even if the number of simulations is held fixed, a Monte Carlo test is able to distinguish between the null hypothesis and alternative hypotheses distant n–  1/2 from the null.
Jackknifing Weighted Least Squares Estimators
Authors: Shao Jun
Published: December 2018

In a heteroscedastic linear regression model, the ordinary jackknife estimator of the asymptotic covariance matrix of the weighted least squares estimator is proved to be inconsistent. A modified jackknife procedure is proposed and shown to produce consistent estimators of the asymptotic covariance matrix. Finite sample performances of the jackknife, the modified jackknife and the customary S method are discussed. Some empirical results are also presented.
Exact Maximum A Posteriori Estimation for Binary Images
Authors: Greig D. M. , Porteous B. T. , Seheult A. H.
Published: December 2018

In this paper, for a degraded two-colour or binary scene, we show how the image with maximum a posteriori  (MAP) probability, the MAP estimate, can be evaluated exactly using efficient variants of the Ford–Fulkerson algorithm for finding the maximum flow in a certain capacitated network. Availability of exact estimates allows an assessment of the performance of simulated annealing and of MAP estimation itself in this restricted setting. Unfortunately, the simple network flow algorithm does not extend in any obvious way to multicolour scenes. However, the results of experiments on two-colour images suggest that, in general, simulated annealing, according to practicable ‘temperature’ schedules, can produce poor approximations to the MAP estimate to which it converges.
Spatial Clustering for Inhomogeneous Populations
Authors: Cuzick Jack , Edwards Robert
Published: December 2018

A new method for detecting spatial clustering of events in populations with non-uniform density is proposed. The method is based on selecting controls from the population at risk and computing interpoint distances for the combined sample. Nonparametric tests are developed which are based on the number of cases among the k nearest neighbours of each case and the number of cases nearer than the k nearest control. The performance of these tests is evaluated analytically and by simulation and the method is applied to a data set on the locations of cases of childhood leukaemia and lymphoma in a defined geographical area. In particular the impact on power of the choice of k and of the ratio of cases to controls is examined. Modifications of the procedure to study distances from predefined objects, to match for known risk factors which would produce unwanted clustering and issues related to estimation are also discussed.
Nonparametric Estimation of Intensities for Sampled Counting Processes
Authors: Guttorp Peter , Thompson Mary Lou
Published: December 2018

Sampled counting processes are often studied using methods for the analysis of stationary time series. We express some time series parameters as smoothed versions of corresponding point process parameters and use these relations to suggest estimates of the point process parameters. In addition, we propose a reconstructive approach to the estimation problem. We derive some properties of the estimators and apply them to data from meteorology, entomology and particle physics. A simulation study indicates which estimators perform better in different situations.
Average Efficiency Factors and Adjusted Orthogonality in Multidimensional Designs
Authors: Park D. K. , Dean A. M.
Published: December 2018

An upper bound is given for the average efficiency factor of a connected multidimensional design. It is shown that, when the design has adjusted orthogonality, its average efficiency factor is a simple function of the average efficiency factors of its component designs.
Disaggregation of Time Series Models
Authors: Wei William W. S. , Stram Daniel O.
Published: December 2018

We develop a model disaggregation method to derive a disaggregate model from a given aggregate model, which is then used to perform data disaggregation. Since a time series model and its autocovariance structure are closely related, we approach the problem by exploring the possibility of estimating the autocovariance structure for the unobserved disaggregated series from the available autocovariances of an aggregate model. Let the time series aggregates be the non-overlapping sums of m consecutive disaggregated observations. Given an aggregate autoregressive integrated moving average ARIMA(p, d, r) model with r ≤ p + d + 1, assume that there is no hidden periodicity of order m. It is shown that, if m is odd or if m is even but all the real roots of the autoregressive polynomial of the given aggregate model are positive, then there exists a disaggregate model whose autocovariances can be uniquely derived from the autocovariances of the given aggregate model. Both non-seasonal and seasonal models are discussed. Empirical examples are presented to illustrate the method.
A Sample Reuse Method for Accurate Parametric Empirical Bayes Confidence Intervals
Authors: Carlin Bradley P. , Gelfand Alan E.
Published: December 2018

Parametric empirical Bayes (EB) methods of point estimation for a vector of unknown parameters date to the landmark paper of James and Stein (1961). The usual approach is to use the mean of the estimated posterior distribution of each parameter, where the estimation of the prior parameters (‘hyperparameters’) is accomplished through the marginal distribution of the data. While point estimates computed this way usually perform well, interval estimates based on the estimated posterior (called ‘naive’ EB intervals) do not. They fail to account for the variability in the estimation of the hyperparameters, generally resulting in subnominal coverage probability in the EB sense defined in Morris (1983a). In this paper we extend our earlier work in which we proposed a conditional bias correction method for developing EB intervals which corrects the deficiencies in the naive intervals. We show how bias correction can be implemented in general via a type III parametric bootstrap procedure, a sample reuse method first employed by Laird and Louis (1987). Theoretical and simulation work indicates that the resulting intervals are accurate with respect to nominal coverage. We give a specific application (to Poisson failure rate data) where we compute simultaneous point and bias-corrected interval estimates.
Jackknifing Linear Estimating Equations: Asymptotic Theory and Applications in Stochastic Processes
Authors: Lele Subhash
Published: December 2018

Let (X  1, X  2, ..., Xn) be a vector of (possibly dependent) random variables having distribution F(X, θ). Let θn be an estimating equation for θ, e.g. the score function or the maximum pseudolikelihood estimating equation in spatial processes. Let θn be the estimator obtained from G such that θn → θ0 in probability and n1/2(θn−θ0)→N(0,V) in distribution. In many situations, it is difficult to derive an analytical expression for V, e.g. for maximum pseudolikelihood estimators for the spatial processes. In this paper, we give a jackknife estimator of V and show that it is weakly consistent. The method consists of deleting one estimating equation (instead of one observation) at a time and thus obtaining the pseudovalues. The method of proof and conditions are similar to those of Reeds with some modifications. The method applies equally to independent and identically distributed random variables, independent but not identically distributed random variables, time- or space-dependent stochastic processes. Our conditions are less severe than Carlstein's who deals with a similar problem of estimating V for dependent observations. We also give some simulation results.
Procrustes Methods in the Statistical Analysis of Shape
Authors: Goodall Colin
Published: December 2018

Two geometrical figures, X and Y, in RK, each consisting of N landmark points, have the same shape if they differ by at most a rotation, a translation and isotropic scaling. This paper presents a model-based Procrustes approach to analysing sets of shapes. With few exceptions, the metric geometry of shape spaces is quite complicated. We develop a basic understanding through the familiar QR and singular value decompositions of multivariate analysis. The strategy underlying the use of Procrustes methods is to work directly with the N x K co-ordinate matrix, while allowing for an arbitrary similarity transformation at all stages of model formulation, estimation and inference. A Gaussian model for landmark data is defined for a single population and generalized to two-sample, analysis-of-variance and regression models. Maximum likelihood estimation is by least squares superimposition of the figures; we describe generalizations of Procrustes techniques to allow non-isotropic errors at and between landmarks. Inference is based on an N x K linear multivariate Procrustes statistic that, in a double-rotated co-ordinate system, is a simple but singular linear transformation of the errors at landmarks. However, the superimposition metric used for fitting, and the model metric, or covariance, used for testing, may not coincide. Estimates of means are consistent for many reasonable choices of superimposition metric. The estimates are efficient (maximum likelihood estimates) when the metrics coincide. F-ratio and Hotelling's R  2-tests for shape differences in one- and two-sample data are derived from the distribution of the Procrustes statistic. The techniques are applied to the shapes associated with hydrocephaly and nutritional differences in young rats.
Identifying Multiple Outliers in Multivariate Data
Authors: Hadi Ali S.
Published: December 2018

We propose a procedure for the detection of multiple outliers in multivariate data. Let X be an n × p data matrix representing n observations on p variates. We first order the n observations, using an appropriately chosen robust measure of outlyingness, then divide the data set into two initial subsets: A ‘basic’ subset which contains p +1 ‘good’ observations and a ‘non-basic’ subset which contains the remaining n - p - 1 observations. Second, we compute the relative distance from each point in the data set to the centre of the basic subset, relative to the (possibly singular) covariance matrix of the basic subset. Third, we rearrange the n observations in ascending order accordingly, then divide the data set into two subsets: A basic subset which contains the first p + 2 observations and a non-basic subset which contains the remaining n - p - 2 observations. This process is repeated until an appropriately chosen stopping criterion is met. The final non-basic subset of observations is declared an outlying subset. The procedure proposed is illustrated and compared with existing methods by using several data sets. The procedure is simple, computationally inexpensive, suitable for automation, computable with widely available software packages, effective in dealing with masking and swamping problems and, most importantly, successful in identifying multivariate outliers.
A Comparison of Variance Estimators in Nonparametric Regression
Authors: Carter C. K. , Eagleson G. K.
Published: December 2018

We compare two estimators of error variance, both based on quadratic forms in the residuals about smoothing spline fits to data. The estimators are compared over the whole range of values of the smoothing parameter as well as for data-based choices of the smoothing parameter. We show that the commonly used estimator of variance has the serious drawback of underestimating the error variance for small choices of the smoothing parameter. This drawback is not shared by a simple, but more computationally intensive, alternative.
Exact and Approximate Posterior Moments for a Normal Location Parameter
Authors: Pericchi L. R. , Smith A. F. M.
Published: December 2018

The forms of first and second posterior moments for a normal location parameter are identified for a rather general class of prior distributions. Exact and approximate illustrations are given where the prior distribution is double exponential or Student t respectively.
Bayesian and Frequentist Bartlett Corrections for Likelihood Ratio and Conditional Likelihood Ratio Tests
Authors: Ghosh J. K. , Mukerjee Rahul
Published: December 2018

This paper characterizes priors under which the Bayesian and frequentist Bartlett corrections for the likelihood ratio and the conditional likelihood ratio (CLR) statistics differ by o(1). It is seen that, except for sample points with negligible probability, the CLR statistic has a posterior distribution for which a posterior Bartlett correction exists. This observation leads to an alternative proof of the existence of a frequentist Bartlett correction for the CLR statistic.
The Identity of Armitage's Two Tests of Heterogeneity of Proportions for Proportional Subclass Numbers
Authors: Gart John J.
Published: December 2018

Armitage discusses two χ2 -tests for heterogeneity of proportions after adjustment for stratification. One is based on a quadratic form of the deviations from expected values and the other is a more easily calculated Pearson χ2 -test. We show, for proportional subclass numbers (or balanced designs), that the two test statistics are identical.
Chaos and Deterministic Versus Stochastic Non-Linear Modelling
Authors: Casdagli Martin
Published: December 2018

An exploratory technique is introduced for investigating how much of the irregularity in an aperiodic time series is due to low dimensional chaotic dynamics, as opposed to stochastic or high dimensional dynamics. Non-linear models are constructed with a variable smoothing parameter which at one extreme defines a non-linear deterministic model, and at the other extreme defines a linear stochastic model. The accuracy of the resulting short-term forecasts as a function of the smoothing parameter reveals much about the underlying dynamics generating the time series. The technique is applied to a variety of experimental and naturally occurring time series data, and the results are compared with dimension calculations.
On Consistent Nonparametric Order Determination and Chaos
Authors: Cheng B. , Tong H.
Published: December 2018

We give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models. We argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos. Setting the stochastic models within the framework of non-linear autoregression, we introduce the notion of a generalized partial autocorrelation and an order. We approach the estimation of the embedding dimension via order determination of an unknown non-linear autoregression by cross-validation, and give justification by proving its consistency under global boundedness. As a by-product, we provide a theoretical justification of the final prediction error approach of Auestad and Tjøstheim. Some illustrations based on the Hénon map and several real data sets are given. The bias of the residual sum of squares as essentially a noise variance estimator is quantified.
Modelling Agent Forecast Distributions
Authors: West Mike
Published: December 2018

Problems in the analysis and interpretation of uncertain inferences obtained from individuals, or agents, are considered in the context of forecasting a scalar random quantity. Based on new theoretical results presented by West and Crosse, Bayesian updating is developed for cases in which agents' opinions are provided in terms of full forecast distributions (discrete or continuous), or partial summaries such as forecast means and variances. A particular focus is on problems in which partial agent information is communicated in terms of a few summary probabilities or, alternatively, quantiles of the agent's distribution. Example models highlight the key features of the theory and illustrate applications.
Spatial Statistics and Bayesian Computation
Authors: Besag Julian , Green Peter J.
Published: December 2018

Markov chain Monte Carlo (MCMC) algorithms, such as the Gibbs sampler, have provided a Bayesian inference machine in image analysis and in other areas of spatial statistics for several years, founded on the pioneering ideas of Ulf Grenander. More recently, the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a Markov random field has opened the way to the use of MCMC in general Bayesian computation. In this paper, we trace the early development of MCMC in Bayesian inference, review some recent computational progress in statistical physics, based on the introduction of auxiliary variables, and discuss its current and future relevance in Bayesian applications. We briefly describe a simple MCMC implementation for the Bayesian analysis of agricultural field experiments, with which we have some practical experience.
Modelling Complexity: Applications of Gibbs Sampling in Medicine
Authors: Gilks W. R. , Clayton D. G. , Spiegelhalter D. J. , Best N. G. , McNeil A. J. , Sharples L. D. , Kirby A. J.
Published: December 2018

We review applications of Gibbs sampling in medicine, involving longitudinal, spatial, covariate measurement and survival models. Applications in immunology, pharmacology, transplantation, cancer screening, industrial epidemiology and genetic epidemiology are discussed.
Estimating Logistic Regression Parameters for Bivariate Binary Data
Authors: McDonald Barry W.
Published: December 2018

Consider bivariate binary data, with possibly different covariates for each marginal binary observation. Suppose that the correlation between paired observations is a nuisance, and the interest lies in estimating logistic regression parameters associated with the covariates. We consider the estimator obtained by using the independence estimating equation, with corrected variance. This estimator can be very efficient and robust compared with maximum likelihood or the estimator of Liang and Zeger. Some comments are made concerning higher dimensional multivariate data.
Modelling Marginal Hazards in Multivariate Failure Time Data
Authors: Liang Kung-Yee , Self Steven G. , Chang Yue-Cune
Published: December 2018

We consider the problem of estimating relative risk parameters in the marginal distribution of failure times from a sample in which individuals are grouped into clusters and failure times for individuals within clusters may be dependent. We assume that the marginal hazard of failure for individuals within a cluster is specified by a relative risk regression model and the structure of dependence between individuals within a cluster is left unspecified. Estimating equations for the regression parameters and an estimator for the cumulative base-line marginal hazard are described. Large sample distribution theory based on a representation of the estimating functions as U-statistics is developed and results from a small simulation study are described. Two examples which originally motivated this work are presented.
Fitting Multivariate Regression Functions by Interaction Spline Models
Authors: Chen Zehua
Published: December 2018

In nonparametric multivariate regression analysis, we seek methods to reduce the dimensionality of the regression function to bypass the difficulty caused by the curse of dimensionality. The original additive models approximate a regression function by additive univariate functions, which overcome successfully the difficulty caused by the curse of dimensionality and have a clear interpretation. But since they ignore any interactions between regressors they are not sufficiently flexible to cope with the case where interactions are of concern. In this paper, we study interactive models in which the regression function is approximated by an analysis-of-variance-type function. The models retain all the advantages of the original additive models but are much more flexible. We study ways of fitting an interactive model by the smoothing spline method. We relate the well-known backfitting algorithm to a typical penalized least squares problem in the context of the smoothing spline method. We develop a non-standard method based on bootstrap methodology for model selection in the interaction spline modelling.
Duality Principle in Order Statistics
Authors: Balasubramanian K. , Balakrishnan N.
Published: December 2018

We establish a duality principle for order statistics in the arbitrary case, using which many known dual results on order statistics can be deduced without a formal proof. The literature on order statistics is rich with such dual pairs derived under different assumptions such as continuity, absolute continuity, exchangeability, etc.
Multiregression Dynamic Models
Authors: Queen Catriona M. , Smith Jim Q.
Published: December 2018

Multiregression dynamic models are defined to preserve certain conditional independence structures over time across a multivariate time series. They are non-Gaussian and yet they can often be updated in closed form. The first two moments of their one-step-ahead forecast distribution can be easily calculated. Furthermore, they can be built to contain all the features of the univariate dynamic linear model and promise more efficient identification of causal structures in a time series than has been possible in the past.
Heterogeneity in Models for Bivariate Survival: The Importance of the Mixing Distribution
Authors: Lindeboom Maarten , Van den Berg Gerard J.
Published: December 2018

We consider bivariate survival models in which the dependence between two survival times is by way of stochastically related unobserved components. We analyse the role of the mixing or heterogeneity distribution in such models and examine how this distribution affects the correlation between the two survival times. We derive explicit expressions for the correlation between the survival times and examine some properties of this correlation. Furthermore we examine to what extent the choice of this heterogeneity distribution is important for the hazards in the sample and their time paths. The time path of the hazard among the survivors is determined by the moments of the distribution describing the heterogeneity among the survivors. We consider particular published mixing distributions. Simulations show that it may be hazardous to estimate bivariate survival models in which the mixing distribution is parameterized univariate, in that a univariate random variable may not be able to account both for the mutual dependence of the survival times and for the change in the composition of the sample over time owing to unobserved heterogeneity.