bluuebunny/crossref_metadata_2025 huggingface dataset:
Efficient Two-Factor Balanced Designs
Authors: Suen Chung-Yi , Chakravarti I. M.
Published: December 2018

Efficient two-factor balanced designs are constructed by using balanced arrays of strength two with indices λ(x, y) = λ0 if x = y or λ1 if x ≠ y. Balanced arrays of this type are constructed from doubly transitive permutation groups and partly resolvable orthogonal arrays. Some two-factor balanced design with all the main effects unconfounded with block effects are constructed.
Maximum Likelihood Estimation for the Near(2) Model
Authors: Smith Richard L.
Published: December 2018

The NEAR(2) model is a model for nonlinear time series with exponential marginals. It formed the subject of a recent discussion paper by Lawrance and Lewis. We explore the possibility of maximum likelihood estimation with particular reference to the difficulties created by discontinuities in the likelihood function. Two exploratory techniques are proposed for resolving those difficulties. This note is an expanded version of a contribution presented orrally at the meeting.
On Some Smoothing Techniques Used in Image Restoration
Authors: Hall Peter , Titterington D. M.
Published: December 2018

The problem is considered of restoring a blurred and/or noisy image using various regularization prescriptions. Preliminary work concerns the invertibility of point spread functions and the construction of a stochastic model for images. A simple linear regularization procedure is introduced, as well as the special cases of constrained deconvolution, maximum entropy restoration and least-squares filtering. Optimal choices for the degree of smoothing are obtained for the case of low noise-to-signal ratios. Certain techniques, prevalent in the image-restoration literature for choosing the degree of smoothing, are shown to oversmooth, in a well-defined sense.
Cross-Validation Shrinkage of Regression Predictors
Authors: Copas J. B.
Published: December 2018

Copas (1983) suggests that shrinkage of predictors can be related to the least squares slope of actual on predicted values in a new set of data. Considering multiple regression in a crossvalidatory setting, this shrinkage slope is approximated by an estimable function of the fitted residuals. Patterns of heteroscedasticity of residual variance can lead to more, or less, shrinkage of least squares. A new predictor using a nonparametric shrinkage multiplier is proposed.
Common Structure of Techniques for Choosing Smoothing Parameters in Regression Problems
Authors: Hall Peter , Titterington D. M.
Published: December 2018

Two general methods have been used for choosing the degree of smoothing in both linear ridge-regression and nonparametric regression. One is based on a criterion of minimum risk and the other is based on a measure of fit to the data as summarised by a statistic based on the residuals. The comparative effects of these two approaches are investigated and one popular version of the second method is shown to oversmooth quite drastically. The work also generates alternative suggestions for data-based smoothing prescriptions, and elucidates the heuristic argument of Wahba (1983).
On Posterior Asymptotic Normality and Asymptotic Normality of Estimators for the Galton-Watson Process
Authors: Scott David
Published: December 2018

For a Galton-Watson process with offspring distribution pθ, where θ is an unknown parameter, asymptotic posterior normality is established for θ and for the mean of the offspring distribution. A form of asymptotic normality for the mean of the offspring distribution is also obtained, without restriction on whether the process is supercritical or not, provided that the total number of offspring increases to infinity. A non-Bayesian procedure is suggested for testing for supercriticality of the Galton-Watson process.
Statistical Models of Chemical Kinetics in Liquids
Authors: Clifford Peter , Green N. J. B. , Pilling M. J.
Published: December 2018

Statistical methods are used increasingly in theoretical chemistry. Applications range from the use of stochastic relaxation techniques in determining minimum energy molecular configurations to dynamical Monte Carlo simulations of molecular motion in liquids. This paper focuses on diffusion-controlled reactions in radiation chemistry. Here the interest is in describing the evolution of isolated clusters, containing a few chemically active particles, resulting from the passage of ionising radiation through a liquid. The subsequent chemistry is determined by the rate at which these particles can encounter each other, pairwise, in the course of their random motion. Classically the trajectories of the particles are described as sample paths of a continuous stochastic process, which in many cases can be assumed to be a diffusion. We have devised an approximate theory based on the approximation that pair distances evolve independently. The effect of this geometric distortion has been studied for small systems by simulation. The extension of this work to a wider class of realistic chemical processes poses many problems. Analytical progress is difficult and methods of simulating large spatial systems are not well developed. Problems which remain to be solved relate to the random geometry of the cluster as its constituent particles diffuse, the development of good approximations to first passage time distributions for diffusions with inhomogeneous drift and methods for the analysis of random motion in a liquid on the basis of detailed non-Markovian models of molecular movement.
Parametric Collapsibility and the Lack of Moderating Effects in Contingency Tables with a Dichotomous Response Variable
Authors: Wermuth Nanny
Published: December 2018

We consider contingency tables having one variable specified as a response with just two categories. We look at conditions for collapsibility of a symmetric and a directed measure of association, the odds-ratio and the relative risk: situations are discussed under which equal partial associations coincide with the corresponding marginal association. Contrary to the odds-ratio the relative risk is collapsible, if there are independencies in the marginal distribution of the influencing variables. This fact is exploited to derive conditions for the lack of a moderating effect, the latter being a much discussed concept in the social science literature.
Does Simple Random Sampling Provide Adequate Balance?
Authors: Cumberland W. G. , Royall R. M.
Published: December 2018

Theoretical arguments imply that, as the sample size increases, simple random sampling does not force balance fast enough to provide robust inference for a finite population total using the ratio estimator. An empirical study shows that under simple random sampling, confidence intervals for the total can have conditional coverage probabilities that are persistently awful as the sample size grows.
Adjusting Belief Structures
Authors: Goldstein Michael
Published: December 2018

Suppose that we have made various belief statements, but not a full probability specification, about a collection of random quantities. How do we modify uncertainties over one subset of the quantities, when given the values of another subset? We suggest an approach in which belief statements are organised into belief structures which are adjusted by observation on related belief structures and derive various properties of such adjustments. We give two examples. In the first example, belief adjustments are related to case statistics for regression diagnostics. In the second example adjusted structures are used to suggest designs for the localised regression model.
Local Computations with Probabilities on Graphical Structures and Their Application to Expert Systems
Authors: Lauritzen S. L. , Spiegelhalter D. J.
Published: December 2018

A causal network is used in a number of areas as a depiction of patterns of ‘influence’ among sets of variables. In expert systems it is common to perform ‘inference’ by means of local computations on such large but sparse networks. In general, non-probabilistic methods are used to handle uncertainty when propagating the effects of evidence, and it has appeared that exact probabilistic methods are not computationally feasible. Motivated by an application in electromyography, we counter this claim by exploiting a range of local representations for the joint probability distribution, combined with topological changes to the original network termed ‘marrying’ and ‘filling-in‘. The resulting structure allows efficient algorithms for transfer between representations, providing rapid absorption and propagation of evidence. The scheme is first illustrated on a small, fictitious but challenging example, and the underlying theory and computational aspects are then discussed.
A Generalization of a Nonparametric Test for Stochastically Ordered Distributions to Censored Survival Data
Authors: Mau Jochen
Published: December 2018

Currently used test statistics for stochastically increasing survival times in a k-sample problem are linear combinations of either Gehan (generalized Wilcoxon) or generalized Savage score sums. The power of these tests depends on a priori knowledge about the true form of the alternative. When isotonic regression is applied to these score sums, it yields a new trend test for right-censored survival data which generalizes to left-truncated survival data and to counting processes. In a Monte Carlo study, its empirical power is at least comparable with that of the linear trend tests in their standard form (without prior information), and it is markedly superior if only the last survival distribution turns out to be stochastically smaller or larger than the control distribution.
Collapsed Two-Way Contingency Tables and the Chi-Square Reduction Principle
Authors: Gilula Zvi , Krieger Abba M.
Published: December 2018

The value of the chi-square statistic for homogeneity of rows and columns in a contingency table remains unchanged if the table is collapsed (grouped) over homogeneous rows and columns. The difference between the chi-square values pertaining to the original table and to the collapsed table (the ‘chi-square reduction‘) is shown to have a meaningful canonical decomposition which can guide the researcher in detecting homogeneous rows and columns. The methodology proposed generalizes an earlier methodology suggested by Gilula for the same problem.
Shape Analysis of Triangles Through Directional Techniques
Authors: Mardia K. V.
Published: December 2018

Kendall has shown how the shape variables of a labelled triangle can be mapped to a point on a sphere. Using Mardia and Dryden's method, we obtain the shape distribution of triangles under Bookstein's model. Various properties are discussed and in particular we show that this distribution, transformed to the sphere, is approximately the Fisher distribution. In particular, we examine the reduction of this distribution when the shape space is reduced to a subset of the sphere. Some practical examples are given, and the hypothesis of central place theory for a geographical data set is reassessed.
A Multivariate Distribution with Weibull Connections
Authors: Crowder Martin
Published: December 2018

A simple form of multivariate distribution is defined which, for certain parameter values, has Weibull marginals. The distribution has a single parameter governing association between the variates, which may be positive, negative or zero. The simple forms for marginal and conditional distributions, hazard functions and densities make it attractive for practical application and interpretation.
Partial Separation in Logistic Discrimination
Authors: Lesaffre E. , Albert A.
Published: December 2018

The problem of maximum likelihood estimates in logistic discrimination is receiving growing attention in the literature. The existence is known to be highly dependent on the data configuration observed. Here we focus on the practically important case where the group samples form a set of clusters that are completely separated from each other, which we call ‘partial separation'. The classical Fisher iris data set is a good example of the problem discussed. A non-existence theorem is proven and a general algorithm is developed to help to distinguish separation from multicollinearity given divergence in maximum likelihood estimation.
Spherical Regression on Matched Pairs of Orientation Statistics
Authors: Prentice Michael J.
Published: December 2018

The matched pairs section of the analysis of 3 times 3 orientation statistics in Prentice (1986) may be improved on by constructing a rotational regression model, an analogue of spherical regression. The results obtained are relatively simple consequences of the work of Chang and are therefore presented here for general orientation statistics—rigid orthogonal m frames in p dimensions—before application to the matched pairs of vectorcardiogram orientations (m = p = 3) which provided the original motivation for this paper.
Chebyshev Polynomial Approximations for Characteristic Function Estimation: Some Theoretical Supplements
Authors: Madan D. B. , Seneta E.
Published: December 2018

Densities for the angle Θ = uX(mod 2π) and the cosine Y = cos(uX) can be expressed, through Fourier methods, in terms of the characteristic function of X. Thus, if X has a characteristic function of closed form, the parameters of its distribution may be estimated by maximum likelihood. The paper addresses the problems of estimability of asymmetric distributions and situations where the method may not apply.
Best Conditional Tests for Separate Families of Hypotheses
Authors: Pace L. , Salvan A.
Published: December 2018

We consider, from the viewpoint of hypotheses testing, the problem of assigning the probability density function of a random variable X to one of two separate families of distributions. Our aim is to study the possibility of constructing best similar tests. For this, we characterize sufficient statistics for the union of the two families; completeness is also studied. We state conditions under which the test statistics are easily obtainable when both families are of the generalized exponential type. As an illustration, many examples are given solving both new and previously worked problems of choice between separate families of hypotheses. The relative merits of the exact best test and of asymptotic tests are empirically investigated in small samples when testing for the log-normal versus the gamma distribution.
Computation of Theoretical Autocovariance Matrices of Multivariate Autoregressive Moving Average Time Series
Authors: Mittnik Stefan
Published: December 2018

Matrix expressions relating the theoretical autocovariances of autoregressive moving average (ARMA) processes to their parameters are derived and used to design an efficient procedure for computing autocovariance sequences of multivariate ARMA processes. The method proposed is more efficient than others suggested in the literature and, in particular, reduces the computational burden associated with exact maximum likelihood estimation of ARMA models. The closed form expressions facilitate the implementation of algorithms for computing multivariate autocovariances.
A New Approach to Frequency Analysis with Amplified Harmonics
Authors: Truong-Van B.
Published: December 2018

For stationary time series having a mixed spectrum, we introduce a new method for estimating the discrete frequencies based on amplification of the sine waves in the time series and on minimization of least squares regressions on amplified waves. Under the assumption that the continuous part of the spectrum arises from a stationary autoregressive moving average process, we show that the estimators for the discrete frequencies obtained with this method are strongly consistent: more precisely, as the serial length n tends to infinity, their bias almost surely converges to zero with the rate n−3/2(logn)δ,δ>12. We also establish a central limit theorem which shows that these estimators have exactly the same asymptotic variance as that given by Whittle's method, which is known to yield the most accurate frequency estimates. However, the new technique has two advantages over Whittle's method. First, it yields an algorithm that is both robust against improper starting estimates and computationally efficient. Secondly, it gives rise to a simple tool based on its amplified harmonics for identifying the number of sine waves in a time series. A simulation study is also reported that gives statistical results on the properties of the new method on simulated series as well as comparative results with Whittle's method.
Continuum Regression: Cross-Validated Sequentially Constructed Prediction Embracing Ordinary Least Squares, Partial Least Squares and Principal Components Regression
Authors: Stone M. , Brooks R. J.
Published: December 2018

The paper addresses the evergreen problem of construction of regressors for use in least squares multiple regression. In the context of a general sequential procedure for doing this, it is shown that, with a particular objective criterion for the construction, the procedures of ordinary least squares and principal components regression occupy the opposite ends of a continuous spectrum, with partial least squares lying in between. There are two adjustable ‘parameters’ controlling the procedure: ‘alpha’, in the continuum [0, 1], and ‘omega’, the number of regressors finally accepted. These control parameters are chosen by cross-validation. The method is illustrated by a range of examples of its application.
Uniform Saddlepoint Approximations and Log-Concave Densities
Authors: Jensen Jens Ledet
Published: December 2018

For a one-dimensional variable with a log-concave density it is shown that the saddlepoint approximations to the density and the distribution function of the mean are uniformly valid. A double-saddlepoint approximation to a conditional distribution function is introduced, which is uniformly valid for a two-dimensional log-concave density. The results are illustrated with the log-normal distribution and the gamma distribution.
Existence of Maximum Likelihood Estimates for Interval-Censored Data from Some Three-Parameter Models with a Shifted Origin
Authors: Nakamura T.
Published: December 2018

Given data in the form of interval-censored observations from a three-parameter distribution, practical criteria are established for the existence of maximum likelihood estimates for three kinds of parameter space. These criteria can be derived by analysing the behaviour of the log-likelihood near a type of boundary of the parameter space, called the probability contents inner boundary. The effectiveness of the criteria is discussed. In the case where the underlying distribution is a three-parameter log-normal distribution, the efficiency of the criteria is evaluated by simulation.
Estimating the Mean and Covariance Structure Nonparametrically When the Data are Curves
Authors: Rice John A. , Silverman B. W.
Published: December 2018

We develop methods for the analysis of a collection of curves which are stochastically modelled as independent realizations of a random function with an unknown mean and covariance structure. We propose a method of estimating the mean function non-parametrically under the assumption that it is smooth. We suggest a variant on the usual form of cross-validation for choosing the degree of smoothing to be employed. This method of cross-validation, which consists of deleting entire sample curves, has the advantage that it does not require that the covariance structure be known or estimated. In the estimation of the covariance structure, we are primarily concerned with models in which the first few eigenfunctions are smooth and the eigenvalues decay rapidly, so that the variability is predominantly of large scale. We propose smooth nonparametric estimates of the eigenfunctions and a suitable method of cross-validation to determine the amount of smoothing. Our methods are applied to data on the gaits of a group of 5-year-old children.
Nonparametric Regression: Optimal Local Bandwidth Choice
Authors: Vieu Philippe
Published: December 2018

Kernel estimators of a regression function are investigated. The bandwidths are locally chosen by a data-driven method based on the minimization of a local cross-validation criterion. This method is shown to be asymptotically optimal with respect to local quadratic measures of errors. Monte Carlo experiments are presented, and finally the method is applied to some data of medical interest.
Exponential Linear Models: A Two-Pass Procedure for Saddlepoint Approximation
Authors: Fraser D. A. S. , Reid N. , Wong A.
Published: December 2018

For an exponential linear model, the saddlepoint method gives accurate approximations for the density of the minimal sufficient statistic or maximum likelihood estimate, and for the corresponding distribution functions. In this paper we describe a simple numerical procedure that constructs such approximations for a real parameter in an exponential linear model, using only a two-pass calculation on the observed likelihood function for the original data. Simple examples of the numerical procedure are discussed, but we take the general accuracy of the saddlepoint procedure as given. An immediate application of this is to exponential family models, where inference for a component of the canonical parameter is to be based on the conditional density of the corresponding component of the sufficient statistic, given the remaining components. This conditional density is also of exponential family form, but its functional form and cumulant-generating function may not be accessible. The procedure is applied to the corresponding likelihood, approximated as the full likelihood divided by an approximate marginal likelihood obtained from Barndorff-Nielsen's formula. A double saddlepoint approximation provides another means of bypassing this difficulty. The computational procedure is also examined as a numerical procedure for obtaining the saddlepoint approximation to the Fourier inversion of a characteristic function. As such it is a two-pass calculation on a table of the cumulant-generating function.
Likelihood and Cost as Path Integrals
Authors: Whittle P.
Published: December 2018

We consider the optimal estimation, prediction and control etc. of time series. The essence of a path integral characterization is considered to be that optimal estimates and decisions etc. are obtained by the free extremization of some form. In Sections 2–5 it is shown for some stock situations that this property is ensured by the direct maximization of likelihood (rather than by appeal to the formally equivalent but operationally contrasting least square principle). In Sections 6–9 a risk-sensitive criterion is taken, in that the criterion is an exponential function of cost rather than cost itself. It is shown that this class of problems admits a natural path integral treatment, in which extremization of stress replaces separate extremization of likelihood and cost. A stochastic maximum principle is derived. For the linear–quadratic–Gaussian (LQG) case, results are exact and depend on proof of a certainty equivalence principle. In the non-LQG case they are valid as large deviation approximations, depending on a formalism going back to Bartlett. Finally, in Sections 10 and 11 we note the efficacy (if not indeed the necessity) of direct likelihood arguments in two other time series areas: recursive estimation and structural models.
A Note on the Construction of Row-And-Column Designs with Two Replicates
Authors: Bailey R. A. , Patterson H. D.
Published: December 2018

This paper shows that useful designs for variety trials with two replicates, each arranged in a compact array of plots, can be derived from existing row-and-column designs for two non-interacting sets of treatments. Combinatorial and statistical properties are preserved; in particular, if the original design is efficient, then so is the new design.
Inferences on Multivariate Measures of Interclass and Intraclass Correlations in Familial Data
Authors: Konishi Sadanori , Khatri C. G. , Rao C. Radhakrishna
Published: December 2018

Multivariate measures of interclass and intraclass correlations are introduced to assess the degree of resemblance between family members with respect to more than one characteristic. Unified estimators for the multivariate measures are proposed as the eigenvalues of certain random matrices constructed by the matrices of the weighted sums of squares and products of observations. Asymptotic distributions of the proposed estimators are derived under the assumption of multivariate normality. The asymptotic results produce approximate confidence intervals and test procedures in a simple way.
Multidimensional Markov Chain Models for Image Textures
Authors: Qian W. , Titterington D. M.
Published: December 2018

Multidimensional Markov chain models are developed for texture. By suitable choice of parameters, textures can be simulated that are similar to those generated by Markov random field models, but the simulation procedure is computationally much more economical. The problem of parameter estimation is examined for non-noisy and noisy data. In the latter case, procedures are developed for simultaneous parameter estimation and image restoration. Illustrative examples are provided.
A Bayesian Decision Theoretic Characterization of Poisson Processes
Authors: Polson Nicholas G. , Roberts Gareth O.
Published: December 2018

A Bayesian decision theoretic methodology to characterize probability mechanisms for point processes is developed with particular reference to mixtures and conditional Poisson processes.
Constrained Monte Carlo Maximum Likelihood for Dependent Data
Authors: Geyer Charles J. , Thompson Elizabeth A.
Published: December 2018

Maximum likelihood estimates (MLEs) in autologistic models and other exponential family models for dependent data can be calculated with Markov chain Monte Carlo methods (the Metropolis algorithm or the Gibbs sampler), which simulate ergodic Markov chains having equilibrium distributions in the model. From one realization of such a Markov chain, a Monte Carlo approximant to the whole likelihood function can be constructed. The parameter value (if any) maximizing this function approximates the MLE. When no parameter point in the model maximizes the likelihood, the MLE in the closure of the exponential family may exist and can be calculated by a two-phase algorithm, first finding the support of the MLE by linear programming and then finding the distribution within the family conditioned on the support by maximizing the likelihood for that family. These methods are illustrated by a constrained autologistic model for DNA fingerprint data. MLEs are compared with maximum pseudolikelihood estimates (MPLEs) and with maximum conditional likelihood estimates (MCLEs), neither of which produce acceptable estimates, the MPLE because it overestimates dependence, and the MCLE because conditioning removes the constraints.
A Goodness-Of-Fit Test for Time Series with Long Range Dependence
Authors: Beran Jan
Published: December 2018

We propose a test statistic for goodness of fit in time series with slowly decaying serial correlations. The asymptotic distribution of the test statistic, originally proposed by Milhoj for time series with smooth spectra, turns out to be the same, under the null hypothesis, even if the spectrum has a pole at 0. In particular, the test is suitable to detect lack of independence in the observations, or estimated residuals, if the first few correlations are small but the decay of the correlations is slow.
Embedded Models in Non-Linear Regression
Authors: Cheng R. C. H. , Evans B. E. , Iles T. C.
Published: December 2018

A problem that is often encountered in non-linear regression is instability in the computation of parameter estimates. Such instability may arise from models with too many parameters, resulting in estimating equations that are ill conditioned and consequent problems of identification. This paper shows that in some cases these difficulties can be explained by the presence of an embedded model, a special case of the original model involving fewer parameters but one that is not readily identified from the parameterization of the model. The paper shows how an embedded model can be identified by making a suitable reparameterization. Moreover this reparameterization removes the problems of numerical instability where an embedded model turns out to be the best fit to the data. The paper suggests how the reparameterization can be chosen systematically. Numerical examples are included to illustrate these points.
On the Expected Amount of Information from a Non-Linear Model
Authors: Polson Nicholas G.
Published: December 2018

An approach to quantifying the amount of information that an experimenter expects to learn from a non-linear model is given. An expected utility for an experiment, ξ, motivated by the asymptotic form of Shannon information gain between prior and posterior, is defined. This leads to a characterization of the experimenter who expects to learn the most from a non-linear model. Such an experimenter has the design-dependent Jeffreys prior. Sufficient regularity conditions for the equivalence with asymptotic Shannon information gain are given. An application to the optimal selection of sample size from a model with exponential family errors and the Michaelis-Menten model is discussed. A link between the regularity conditions for asymptotic posterior normality and the Jeffreys prior is given.
A Type of Second-Order Asymptotic Independence
Authors: Barndorff-Nielsen O. E. , Blæsild P.
Published: December 2018

Suppose that a random vector x is partitioned into two subvectors and that generic coordinates of these subvectors are indicated by xa, xb, … and xr, xs, … respectively. If the joint cumulant (matrix) (ka,r) equals 0 then xa and xr are asymptotically independent to order O(n  -1/2) under ordinary repeated sampling. When the third-order joint cumulant (array) (ka,b,r) is also 0 this independence statement may be sharpened to asymptotic independence to order O(n  -1) of xr and the residuals of xa after quadratic regression on xr. Various examples where this applies to likelihood analysis are discussed.
Multivariate Regression Analyses for Categorical Data
Authors: Liang Kung-Yee , Zeger Scott L. , Qaqish Bahjat
Published: December 2018

It is common to observe a vector of discrete and/or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes. The response vector can comprise repeated observations on one variable, as in longitudinal studies or genetic studies of families, or can include observations for different variables. This paper discusses a class of models for the marginal expectations of each response and for pairwise associations. The marginal models are contrasted with log-linear models. Two generalized estimating equation approaches are compared for parameter estimation. The first focuses on the regression parameters; the second simultaneously estimates the regression and association parameters. The robustness and efficiency of each is discussed. The methods are illustrated with analyses of two data sets from public health research.
Diagnostic Results Concerning a Measure of Rotatability
Authors: Khuri André I.
Published: December 2018

The measure of rotatability introduced by Khuri is utilized to determine the closeness of the prediction variance to being constant on a sphere S(r) of radius r centred at the origin. In particular, several upper bounds on the range of the prediction variances on S(r) are obtained. These upper bounds provide information about the amount of dispersion that can exist in the prediction variances on all spheres inside a region of interest. Such information gives added insight into the nature of ‘near rotatability’ of a given response surface design. This is illustrated in an example involving four designs: the 32 factorial and three hybrid designs described by Roquemore, namely the 310, 311A and 311B designs.
Likelihood, Quasi-Likelihood and Pseudolikelihood: Some Comparisons
Authors: Nelder J. A. , Lee Y.
Published: December 2018

There is considerable interest in the fitting of models jointly to the mean and dispersion of a response. For the mean parameter, the Wedderburn estimating equations are widely accepted. However, there is some controversy about estimating the dispersion parameters. Finite sampling properties of several dispersion estimators are investigated for three models by simulation. We compare the maximum extended quasi-likelihood estimator, the maximum pseudolikelihood estimator and the maximum likelihood estimator, if it exists. Of these estimators, the maximum extended quasi-likelihood estimator is usually superior in minimizing the mean-squared error.
Estimating Dimension in Noisy Chaotic Time Series
Authors: Smith Richard L.
Published: December 2018

This paper is concerned with estimating the dimension of an attractor. The Grassberger–Procaccia procedure for estimating the correlation dimension is reviewed, and some alternative estimators considered. A particularly important problem is to estimate dimension in the presence of noise in the data. In pursuit of this, a modification of the Grassberger–Procaccia procedure is proposed, and illustrated on both artificial and real data sets.
Finding Chaos in Noisy Systems
Authors: Nychka Douglas , Ellner Stephen , Gallant A. Ronald , McCaffrey Daniel
Published: December 2018

In the past 20 years there has been much interest in the physical and biological sciences in non‐linear dynamical systems that appear to have random, unpredictable behaviour. One important parameter of a dynamical system is the dominant Lyapunov exponent (LE). When the behaviour of the system is compared for two similar initial conditions, this exponent is related to the rate at which the subsequent trajectories diverge. A bounded system with a positive LE is one operational definition of chaotic behaviour. Most methods for determining the LE have assumed thousands of observations generated from carefully controlled physical experiments. Less attention has been given to estimating the LE for biological and economic systems that are subjected to random perturbations and observed over a limited amount of time. Using nonparametric regression techniques (neural networks and thin plate splines) it is possible to estimate the LE consistently. The properties of these methods have been studied with simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (1820–1900). On the basis of a nonparametric analysis there is little evidence for low dimensional chaos in these data. Although these methods appear to work well for systems perturbed by small amounts of noise, finding chaos in a system with a significant stochastic component may be difficult.
A Central Limit Theorem for Latin Hypercube Sampling
Authors: Owen Art B.
Published: December 2018

Latin hypercube sampling (LHS) is a technique for Monte Carlo integration, due to McKay, Conover and Beckman. M. Stein proved that LHS integrals have smaller variance than independent and identically distributed Monte Carlo integration, the extent of the variance reduction depending on the extent to which the integrand is additive. We extend Stein's work to prove a central limit theorem. Variance estimation methods for nonparametric regression can be adapted to provide N  1/2-consistent estimates of the asymptotic variance in LHS. Moreover the skewness can be estimated at this rate. The variance reduction may be explained in terms of certain control variates that cannot be directly measured. We also show how to combine control variates with LHS. Finally we show how these results lead to a frequentist approach to computer experimentation.
Information Bounds for the Conditional Hazard Ratio in a Nested Family of Regression Models
Authors: Sasieni Peter
Published: December 2018

Information bounds are obtained for a parametric treatment effect in a nested family of models for the conditional distribution of survival given confounding variables. The family extends from the fully parametric Weibull model to a completely nonparametric model for right-censored data and includes the Cox model and the stratified version of it. It is shown that, in certain special cases, there is no loss of information about the treatment effect when going from a fully parametric to a nonparametric specification of the conditional hazard function.
Maximum Likelihood Regression on Relevant Components
Authors: Helland Inge S.
Published: December 2018

Most regression methods for multicollinear data are only based on assumptions about the conditional distribution of the dependent variable, given the explanatory variables. Here we propose a new prediction method taking a joint multinomial distribution for all the variables as the starting point. A hypothesis is formulated stating that only a fixed small number of components constructed from the explanatory variables is relevant for prediction, and maximum-likelihood-type estimates of the parameters in the model are developed under this hypothesis.
Time Series of Continuous Proportions
Authors: Grunwald Gary K. , Raftery Adrian E. , Guttorp Peter
Published: December 2018

A vector of continuous proportions consists of the proportions of some total accounted for by its constituent components. An example is the proportions of world motor vehicle production by Japan, the USA and all other countries. We consider the situation where time series data are available and where interest focuses on the proportions rather than the actual amounts. Reasons for analysing such times series include estimation of the underlying trend, estimation of the effect of covariates and interventions, and forecasting. We develop a state space model for time series of continuous proportions. Conditionally on the unobserved state, the observations are assumed to follow the Dirichlet distribution, often considered to be the most natural distribution on the simplex. The state follows the Dirichlet conjugate distribution which is introduced here. Thus the model, although based on the Dirichlet distribution, does not have its restrictive independence properties. Covariates, trends, seasonality and interventions may be incorporated in a natural way. The model has worked well when applied to several examples, and we illustrate with components of world motor vehicle production.
Spherical Median Axes
Authors: Fisher N. I. , Lunn A. D. , Davies S. J.
Published: December 2018

In an earlier paper, Fisher defined the notion of a median direction for data from a unimodal distribution of three-dimensional unit vectors and studied its statistical properties. Corresponding notions of a median axis for bipolar axial or great circle data were also suggested but not analysed. This paper gives a statistical treatment of spherical median axes, including their asymptotic relative efficiencies, and a comparison with the customary principal or polar axis estimators using asymptotic relative efficiencies. The sample spherical median axes enjoy the same sort of resistance to outlying values as do the median vector, spatial median and linear median.
Detecting Heteroscedasticity in Nonparametric Regression
Authors: Eubank R. L. , Thomas Will
Published: December 2018

Diagnostic tests and plots are proposed for detecting heteroscedasticity in nonparametric regression. The large and small sample power properties are studied for a class of test statistics for the hypothesis of homogeneous variances. New diagnostic plots are also developed and illustrated.
Saddlepoint Approximations for Tests of Block Independence, Sphericity and Equal Variances and Covariances
Authors: Butler Ronald W. , Huzurbazar S. , Booth James G.
Published: December 2018

Approximations for the null distribution of various test statistics arising in multivariate analysis using Skovgaard's saddlepoint formula are discussed. In each case an approximation is found with such accuracy as to obviate the need for exact computation.
Valid Restricted Randomization for Unbalanced Designs
Authors: Monod Hervé , Bailey R. A.
Published: December 2018

Sufficient conditions for strong validity of randomization are given, for designs whose block structure corresponds to an association scheme. These conditions generalize previous results for orthogonal designs and balanced designs. Applications to restricted randomization of neighbour-balanced designs are discussed.
Simple Modifications for Signed Roots of Likelihood Ratio Statistics
Authors: Diciccio Thomas J. , Martin Michael A.
Published: December 2018

In the context of inference about a scalar parameter in the presence of nuisance parameters, some simple modifications for the signed root of the log-likelihood ratio statistic R are developed that reduce the order of error in the standard normal approximation to the distribution of R from O(n  -1/2) to O(n  -1). Barndorff-Nielsen has introduced a variable U such that the error in the standard normal approximation to the distribution of R + R  -1 log(U/R) is of order O(n  -3/2), but calculation of U requires the specification of an exact or approximate ancillary statistic A. This paper proposes an alternative variable to U, denoted by T, that is available without knowledge of A and satisfies T = U + Op(n  -1) in general. Thus the standard normal approximation to the distribution of R + R  -1 log(T/R) has error of order O(n  -1), and it can be used to construct approximate confidence limits having coverage error of order O(n  -1). In certain cases, however, T and U are identical. The derivation of T involves the Bayesian approach to constructing confidence limits considered by Welch and Peers, and Peers. Similar modifications for the signed root of the conditional likelihood ratio statistic are also developed, and these modifications are seen to be useful when a large number of nuisance parameters are present. Several examples are presented, including inference for natural parameters in exponential models and inference about location-scale models with type II censoring. In each case, the relationship between T and U is discussed. Numerical examples are also given, including inference for regression models, inference about the means of log-normal distributions and inference for exponential lifetime models with type I censoring, where Barndorff-Nielsen's variable U is not available.
Correlation Tests for Non-Linear Alternatives
Authors: Barbour A. D. , Cost M.
Published: December 2018

A new class of statistics for detecting association between two variables is proposed. The statistics improve on traditional correlation coefficients by being sensitive to U-shaped and other non-linear relationships between the variables. If both variables are one dimensional, they can be simply defined in terms of ranks, so that their null distributions can in principle be tabulated. Asymptotic discrimination of order n−  1/2 is obtained for any alternative of a continuous functional relationship between the variables. Power against trend alternatives is investigated by simulation.
Strong Collapsibility of Association Measures in Linear Models
Authors: Geng Zhi , Asano Chooichiro
Published: December 2018

A measure of association in linear models is strongly collapsible over a discrete background variable if it remains unchanged no matter how the background variable is partially pooled and if it also coincides with the corresponding marginal measure of association. Strong collapsibility implies that the measure of association can be studied no matter how the background variable is categorized and no matter whether or not it is recorded. In this paper, necessary and sufficient conditions are given for strong collapsibility of the measure of association in linear models.
Varying-Coefficient Models
Authors: Hastie Trevor , Tibshirani Robert
Published: December 2018

We explore a class of regression and generalized regression models in which the coefficients are allowed to vary as smooth functions of other variables. General algorithms are presented for estimating the models flexibly and some examples are given. This class of models ties together generalized additive models and dynamic generalized linear models into one common framework. When applied to the proportional hazards model for survival data, this approach provides a new way of modelling departures from the proportional hazards assumption.
Regional Modelling of Extreme Storms Via Max-Stable Processes
Authors: Coles Stuart G.
Published: December 2018

Asymptotic models for extremes of random processes often form the basis for estimating the extremal behaviour of environmental phenomena. Most such phenomena have a spatial dimension, and the aim of this paper is to develop a procedure for modelling in continuous space the spatial dependence within extreme events. A principal objective in the analysis—as with other current research on extremes—is to base inference on as much of the available data as possible. The modelling procedures are justified on simulated data and subsequently applied to a series of rainfall data.
Improved Estimators of Variance Components with Smaller Probability of Negativity
Authors: Kelly Robert J. , Mathew Thomas
Published: December 2018

A linear model with two variance components is considered, one variance component (say, σ  1  2 ≥ 0) corresponding to a random effect, and a second variance component (say, σ  2 > 0) corresponding to the experimental errors. A class of invariant quadratic estimators (IQEs) is characterized, having uniformly smaller mean-squared error (MSE), and uniformly smaller probability of negativity, compared with the analysis-of-variance (ANOVA) estimator of σ  1  2. It turns out that for balanced models, among IQEs of σ  1  2 with uniformly smaller MSE than its ANOVA estimator, there exists an IQE with the smallest probability of being negative, uniformly in the parameters. The results are applied to the balanced one-way classification model. Numerical computations show that the MSE improvement achievable through the use of the proposed estimators is quite significant. It is noted that, for the nonnegative estimation of σ  1  2, truncation of one of the proposed estimators at 0 provides a satisfactory solution. Extension of the results to the unbalanced case, and to the general mixed model with balanced data, is also indicated.
An Improved Randomized Response Strategy
Authors: Mangat N. S.
Published: December 2018

A randomized response procedure is proposed which has the benefit of simplicity over that of Mangat and Singh. Conditions are obtained under which the proposed strategy is better than those of Warner or Mangat and Singh.
Affine Invariant Multivariate One-Sample Sign Tests
Authors: Hettmansperger Thomas P. , Nyblom Jukka , Oja Hannu
Published: December 2018

Brown and Hettmansperger introduced an affine invariant bivariate analogue of the sign test based on the generalized median of Oja. In this paper its general multivariate extension is presented. The proposed multivariate permutation or sign change test is affine invariant with an easily computable covariance matrix. For elliptic distributions the proposed test and Randles's test are asymptotically equivalent. Formulae for calculating asymptotic relative efficiencies of the proposed test and the Oja multivariate median are given. Lower bounds for the efficiencies of the new test (and the Randles test) relative to the classical Hotelling test are established for elliptic alternatives and for unimodal elliptic alternatives. Relative efficiencies under multivariate t-distributions are also tabulated. The theory is illustrated by a simple example.
Nonparametric Estimation of Component Lifetime Based on Masked System Life Test Data
Authors: Schäbe Hendrik
Published: December 2018

In this paper we study the model of masked data sets, i.e. data sets consisting of failure times of a system and sets of failed components. Failing components can yield further failures. The initially failed component is unknown. Initial failures and later failures are indistinguishable. A nonparametric estimator of the component lifetime distribution is derived. We prove weak convergence of the estimator to a Gaussian process.