bluuebunny/crossref_metadata_2025 huggingface dataset:
Factorial Experiments in Generalized Cyclic Row-Column Designs
Authors: John J. A. , Lewis S. M.
Published: December 2018

Factorial experiments can be set out in row-column designs using the generalized cyclic method of construction. From parallel results on block designs, it is shown how the confounding schemes and efficiency factors of such designs are obtained. Simple rules for constructing useful designs are given.
Likelihood Based Inference Using Signed Ranks for Matched Pairs
Authors: Pettitt A. N.
Published: December 2018

Exact and approximate inference based on the marginal likelihood which results from the signed ranks of the within pair differences of independent matched pairs is considered. Inference is made using Bayesian ideas. A quickly computed approximate analysis is introduced and this is shown to be extremely good when the within pair differences are assumed to have a normal distribution. Numerical comparisons are also made when the differences have a logistic distribution, but for this case the approximation is not as good. The approximations involve the Wilcoxon signed rank statistic and the half-normal scores signed rank statistic. The approximation is extended to consider regression models for matched pairs data. An application is given illustrating the ideas.
On the Hypergeometric Birth Process and Some Implications About the Gamma Distribution Representation
Authors: Ferreri Carlo
Published: December 2018

A stochastic birth process {X  H(t), t ≥ 0} is derived from a special parametric form of the probability generating function (p.g.f.) of the four-parameter hypergeometric series distribution (HSD). In such context the generalized Waring distribution is obtained as a special equilibrium distribution. The distribution of XH(t) is expressed as a compound distribution whose compounder is a generally time-dependent hyper-gamma distribution (HGD). This compounder is, in turn, regarded as a convolution of a gamma distribution (GD) and a generalized HSD distribution whose generalizer is an exponential distribution (ED). This allows, for example, a separation of the effects of proneness and liability when the negative binomial process is assumed in accident theory. The result is extended by showing that the gamma distribution can be represented as an infinite convolution of generalized negative binomial distributions having ED's as generalizers. The HGD is similarly expressed.
Inequalities on Rank Correlation with Missing Data
Authors: Papaioannou Takis , Loukas Sotiris
Published: December 2018

New inequalities are obtained for Spearman's footrule, its associated sum of the absolute values of rank differences, the number of discordant pairs associated with Kendall's tau and the coefficient of concordance of several rankings when one or more individuals are missing or added to the data.
A Note on Conditional Prediction in the Multivariate Linear Model
Authors: Reinsel Gregory
Published: December 2018

In this paper we investigate the effects of errors in the estimation of the covariance structure of a multivariate random vector on the prediction of unobserved components of a further vector conditional on the values of the remaining observed components of the vector. We derive an exact expression for the increase in the mean square error of the usual least squares predictor due to these estimation errors which shows that the increase is substantial for small samples.
An Optimal Property of Kernel Estimators of a Probability Density
Authors: Hall Peter
Published: December 2018

Let f̂(x) be an asymptotically unbiased, nonparametric estimator of the density f(x) based on observations falling within the interval (x – ∈, x + ∈), where ∈ → 0 as the sample size diverges. We derive a lower bound to the variance of f(x), and prove that the bound is attained by a kernel estimator. This result is extended to estimates of f  (r)(x), and to estimates of derivatives whose bias is improved by taking out the effects of unwanted terms.
On Extensions of the Gauss-Markov Theorem to the Case of Stochastic Regression Coefficients
Authors: Pfeffermann D.
Published: December 2018

Several theorems which extend the Gauss-Markov theorem to the case of stochastic vectors of coefficients are presented in the literature. Some of these theorems, although differently formulated and proved, are based on the same assumptions and arrive at similar results. Some theorems are special cases of theorems, already existing in the literature. In other cases, theorems, which appear to be similar in their underlying assumptions and conclusions, differ in ways that are not immediately apparent. A new extension, optimal prediction of coefficients of unobserved relations, is considered.
Stochastic Models for Rock Instability in Tunnels
Authors: McCullagh Peter , Lang Peter
Published: December 2018

We consider a tunnel of rectangular cross-section driven in hard rock containing three joint sets (series of parallel discontinuity planes that traverse the rock mass). If three planes intersect at a point just above the tunnel, a tetrahedron is formed whose base appears as a triangle in the tunnel roof. If the base triangle is contained entirely between the tunnel walls, the tetrahedral block is said to be kinematically unstable and may require support. Overlapping tetrahedra may also occur. In this paper we begin with the simplifying assumption that the spacings within each joint set are exponentially distributed, a model previously considered by Lang (1979), Warburton (1980), Pahl (1981), and consistent with the empirical findings of Priest and Hudson (1976). The principal results concern the total volume and base area of the overbreak (kinematically unstable rock), the distribution of apex height and the number of apexes, together with the number of simple blocks, composite blocks and key blocks. A similar model based on three independent gamma processes is developed and shows that increased regularity of the spacings leads to an increase in the volume of overbreak. Finally, it is shown how some of the results may be applied to circular and semi-circular tunnels.
Poisson Convergence for Dissociated Statistics
Authors: Barbour A. D. , Eagleson G. K.
Published: December 2018

A Poisson limit theorem is derived for the number of “large” values observed among comparisons of independent, but not necessarily identically distributed random variables. The comparisons made need not be the same and may depend on the two variables being compared. An application to the assessment of large numbers of correlation coefficients is given.
Asymptotic Behaviour of Conditional Maximum Likelihood Estimators in a Certain Exponential Model
Authors: Bar-Lev Shaul K.
Published: December 2018

In this paper we investigate the asymptotic behaviour of the conditional maximum likelihood estimator in an exponential model of order 2 introduced in Bar-Lev and Reiser (1982). This model, with natural parameters θ1 and θ2 (the latter a nuisance parameter), is characterized by expressing θ2 as a particular function of the mixed parameters. A direct proof is given for the asymptotic equivalence of the conditional maximum likelihood estimator and the maximum likelihood estimator for θ1. Some exact results concerning the relationships between these estimators are also presented.
Sequential Treatment Allocation Using Biased Coin Designs
Authors: Smith Richard L.
Published: December 2018

Biased coin designs are used in clinical trials as a means of achieving approximate balance within a randomized design. In this paper, detailed results are obtained for a class of biased coin designs from the point of view of the degree of balance achieved, selection bias, accidental bias and inference following the randomized trial. These results are developed in detail for the simple situation with two treatments and no prognostic factors. They are then extended to cover (i) more than two treatments with no prognostic factors, (ii) two treatments with prognostic factors.
Variance Estimation for the Combined Ratio and Combined Regression Estimators
Authors: Wu C. F. Jeff
Published: December 2018

A class of variance estimators including the standard ones, for the combined ratio estimator and for the combined regression estimator, is proposed. Optimal variance estimators in terms of minimizing the mean square error are obtained. Both model-free and model-dependent comparison of some standard variance estimators is made. For the combined ratio estimator, a new formula for the second order expansion of its mean square error is derived; the jackknife variance estimator is shown to be closely approximated by a (computationally) simpler estimator in the proposed class; and both model-free and model-dependent comparison of the biases of the variance estimators in the proposed class is made.
On Maximizing Missing Information About a Hypothesis
Authors: Eaves D. M.
Published: December 2018

The Bayesian test of a normal mean H  0 : μ = μ0 is revisited. Several interpretations of prior indifference are contrasted; each employs a prior probability for H  0 which in some sense maximizes the information – negative entropy – missing from the prior.
A Note on the Analysis of Resolvable Block Designs
Authors: Speed T. P. , Williams E. R. , Patterson H. D.
Published: December 2018

Yates (1939, 1940) pointed out that ordinary randomized block methods can be used, if necessary, to analyse lattice designs. This useful property is shown to extend to any resolvable incomplete block design including designs with blocks of unequal size.
Testing for Serial Correlation in Regression with Missing Observations
Authors: Robinson P. M.
Published: December 2018

In order to test for serial correlation in residuals for static time series regression in the presence of missing data, the score principle is applied both to the likelihood conditional on the observation times, and to an unconditional form of likelihood. Asymptotic distributions of the test statistics are established, under both the null hypothesis of no serial correlation, and sequences of local, correlated, alternatives, enabling analytic comparison of efficiency.
Preservation of Optimality Under Loss of Varieties
Authors: Constantine Gregory M.
Published: December 2018

Designs most resistant to loss of varieties are found in certain settings. It is shown that a design is most resistant to loss of up to t – 2 varieties if and only if it is a t-design. When loss of both varieties and the corresponding blocks is likely to occur, most resistant designs turn out to be nonbinary.
The Minimax Risk for Two-Stage Procedures in Clinical Trials
Authors: Bather John , Simons Gordon
Published: December 2018

The risk involved in a trial to compare two medical treatments is shared by patients who receive the inferior treatment during the experimental phase and those remaining after the experiment who might all receive the inferior treatment if the results are misleading. We consider the maximum of this risk with respect to the unknown probabilities of success and seek allocation rules that minimize this quantity, for a given total of patients. It needs extensive computations to find such minimax procedures, but there are simple and almost equally effective allocation rules based on a truncated sequential probability ratio test.
A Non-Gaussian State Space Model and Application to Prediction of Records
Authors: Smith R. L. , Miller J. E.
Published: December 2018

We develop a class of state space models for censored data. The basic model assumes an exponential distribution for the observations, conditionally on unobserved state variables. The model may be generalised by allowing transformations. We develop an application to the prediction of records. This is illustrated with some athletics data, though we also discuss briefly the possibility of more general applications connected with extreme values.
Design Effects of Two-Stage Sampling
Authors: Skinner C. J.
Published: December 2018

The effect of a two-stage sampling design on statistical inference is discussed. A definition of a design effect is given. The structure of design effects for a class of statistics is investigated. Results have both a design-based and a model-based interpretation. The relation between design effects for multivariate statistics and design effects for univariate statistics is considered.
Estimates for a Regression Parameter Using Ranks
Authors: Pettitt A. N.
Published: December 2018

In this paper we consider the estimation of a regression parameter in a partially specified model where, after an unknown monotone increasing transformation, the response is normally distributed and related to an explanatory variable by a linear model. We consider a Monte Carlo resampling technique due to Doksum and various modifications of it. In addition a simple estimate based on a modification of Fisher and Yates' normal scores method and an estimate using the means of ranks are considered. The estimates are compared using simulation.
On Circuits and Optimality Conjectures for Block Designs
Authors: Wild Peter
Published: December 2018

We show that Conjecture 5 of John and Williams (1982) implies a strong condition on the numbers of circuits in the variety concurrence graph of an A-optimal binary design. This is related to a conjecture of Paterson (1983) concerning circuits and A-optimality.
Corrected Maximum Likelihood in Non-Regular Problems
Authors: Cheng R. C. H. , Iles T. C.
Published: December 2018

Maximum likelihood (ML) estimation can break down for distributions like the three-parameter Weibull or gamma where one of the parameters is a limit of the range of the distribution. It is suggested that use of the product of densities for the likelihood function is a misapplication of the ML method in such non-regular cases, and that this is the reason for its failure. It is shown that ML works quite satisfactorily if the likelihood is corrected in an appropriate way.
Nonparametric Analysis of Covariance with Ordered Alternatives
Authors: Marcus Ruth , Genizi Abraham
Published: December 2018

The problem of comparing k populations with respect to a univariate response variable Y against an ordered alternative is considered when a covariate X is taken into account using the matched-observations method of Quade (1982). Two test statistics are considered: one is Quade's statistic generalized for arbitrary k and the second is the Terpstra-Jonckheere statistic generalized for grouped data. Some simulations are carried out to study their small sample performance for different distributions of Y and X and for different sizes of the matched-observations classes. Finally, a general method of devising stepwise multiple testing procedures with fixed experiment wise error is applied to our problem of nonparametric analysis of covariance with ordered alternatives. An example of a neighbouring plots model is reanalyzed.
The Efficiency Factors of Particular Effects in Designs with Orthogonal Factorial Structure
Authors: Paterson L. J.
Published: December 2018

We investigate partial confounding which is not necessarily orthogonal to blocks. The efficiency factor of a main effect or interaction is related to the numbers of paths of a certain type in the treatment concurrence graph. From this we can derive combinatorial principles that help find designs with large efficiency factor for the effect or interaction. The principles are heuristically cogent and convenient, and are a natural generalisation of existing practice. Some illustration is given of their value in finding good generalised-cyclic designs.
On the Optimality of Some Tests of the Error Covariance Matrix in the Linear Regression Model
Authors: Honda Yuzo
Published: December 2018

This paper adopts the approach of conditional distributions to investigate the optimality of some tests of the error covariance matrix in the linear regression model. Specifically, we show that point optimal tests, advocated by King and Evans and King, are the most powerful similar test. We also derive the locally best similar and the locally best unbiased similar tests. Finding the latter cumbersome to apply, we then propose the asymptotically best similar and the asymptotically best unbiased similar (ABUS) tests as alternative criteria. We show that the two-sided Durbin–Watson test is ABUS against serial correlation and that the two-sided Lagrange multiplier test is ABUS against heteroscedastic disturbances.
Recursive Estimation of Autoregressions
Authors: Hannan E. J. , McDougall A. J. , Poskitt D. S.
Published: December 2018

This paper is concerned with the recursive estimation of autoregressive models, in particular the realtime determination of the order of such a specification. A model selection criterion based on the minimum description length principle due to Rissanen is discussed and strong consistency in the stationary situation is shown. Alternative criteria are also considered and modifications required to introduce forgetting when the procedures are implemented in the non-stationary case are presented. Some simulation evidence on the performance of the criteria when applied to stationary and non-stationary processes is given.
Median Estimation in the Presence of Auxiliary Information
Authors: Kuk Anthony Y. C. , Mak T. K.
Published: December 2018

This paper deals with the estimation, under simple random sampling, of a finite population median in the presence of an auxiliary variable. An estimator which is a simple modification of the ratio estimator for the mean is studied along with two other proposed estimators derived from a different approach. These estimators are applicable in situations where only the population median or a grouped frequency distribution of the auxiliary variable is known. On the basis of the asymptotic properties derived and some simulation results, the efficiencies of these estimators are compared. It is shown that the two estimators proposed, unlike that obtained from modifying the ratio estimator, always dominate the sample median in mean-squared error and that their efficiencies depend directly on the probability of ‘concordance’ rather than on the validity of a linearity assumption between the survey variable and the auxiliary variable. When the linearity assumption is approximately satisfied, the three estimators are seen to have comparable efficiencies, while the modified ratio estimator is found to be quite sensitive to departures from the linearity assumption.
On Substantive Research Hypotheses, Conditional Independence Graphs and Graphical Chain Models
Authors: Wermuth Nanny , Lauritzen Steffen Lilholt
Published: December 2018

Graphs consisting of points, and lines or arrows as connections between selected pairs of points, are used to formulate hypotheses about relations between variables. Points stand for variables, connections represent associations. When a missing connection is interpreted as a conditional independence, the graph characterizes a conditional independence structure as well. Statistical models, called graphical chain models, correspond to special types of graphs which are interpreted in this fashion. Examples are used to illustrate how conditional independences are reflected in summary statistics derived from the models and how the graphs help to identify analogies and equivalences between different models. Graphical chain models are shown to provide a unifying concept for many statistical techniques that in the past have proven to be useful in analyses of data. They also provide tools for new types of analysis.
Score Tests in Generalized Linear Measurement Error Models
Authors: Stefanski Leonard A. , Carroll Raymond J.
Published: December 2018

Hypothesis tests in generalized linear models are studied under the condition that a surrogate w is observed in place of the true predictor x. The efficient score test for the hypothesis of no association depends on the conditional expectation E(x|w) which is generally unknown. The usual test substitutes w for E(x|w) and is asymptotically valid but not efficient. We investigate two new test statistics appropriate when w = x + z where z is an independent measurement error. The first is a Wald test based on estimators corrected for measurement error. Despite the correction for attenuation in the estimator, this test has the same local power as the usual test. The second test employs an estimator of E(x|w) and is both asymptotically efficient for normal errors and approximately efficient when the measurement error variance is small.
Strata for Randomized Experiments
Authors: Bailey R. A.
Published: December 2018

Fisher argued that, for completely randomized designs, block designs and Latin square designs, his proposed randomization justified his proposed analysis of variance in that the latter gives valid estimates of experimental error even under very weak assumptions about the plot effects. In the succeeding 50 years a large number of more complicated experimental structures has been proposed and used. Randomization of these structures is rarely discussed in more than a cursory way, and its relationship with the analysis is mentioned even less frequently. Here I show that Fisher's ideas on validity and randomization may be set in a symmetry framework. Randomization is done by permuting the plot names, and that randomization procedure is used to justify the assumption of a relatively simple model for plot effects. Permutations are chosen at random from a specified permutation group. Except in the simplest cases, determination of the strata for the analysis of variance needs some knowledge of group theory. The relevant results from group theory are presented, together with some straightforward techniques which depend on them. The strata are then found for many large classes of examples, including the majority of experimental structures in current use.
Posterior Bayes Factors
Authors: Aitkin Murray
Published: December 2018

A general procedure for computing Bayes factors for the comparison of arbitrary models is described, based on the use of the posterior mean of the likelihood under each model rather than the usual prior mean. The use of the posterior mean has several advantages, including reduced sensitivity to variations in the prior and the avoidance of the Lindley paradox in testing point null hypotheses. The frequency properties of the new procedure are evaluated in standard examples, and a non-standard example is analysed to show the considerable differences possible between prior and posterior means of the likelihood. Several different justifications of the procedure are given, and a non-Bayesian direct likelihood interpretation is described.
Grouped Likelihood for the Shifted Power Transformation
Authors: Atkinson Anthony C. , Pericchi Luis R. , Smith Richard L.
Published: December 2018

Estimation of parameters is considered in the shifted power transformation family of models. The ordinary likelihood function is shown to behave poorly, often having no local maximum. It is argued that a grouped likelihood approach avoids these difficulties, and the practical implementation of this approach is considered. Bayesian inference is also considered, though more briefly, and it is argued that there are again difficulties with an ungrouped likelihood approach. The results provide further evidence of the utility of grouped likelihood approaches to inference in non-regular problems.
On the Relationship between Bayesian and Non-Bayesian Interval Estimates
Authors: Severini Thomas A.
Published: December 2018

Let Y  1, ..., Y  n denote independent observations each distributed according to a density depending on a scalar parameter θ. Suppose that we are interested in constructing an interval estimate for 0. One approach is to construct a confidence region with a specified coverage probability based on the likelihood ratio test statistic. Another approach is to construct a highest posterior density region with a specified posterior probability. The goal of this paper is to study the relationship between these two approaches. In particular, we derive asymptotic expansions for the posterior probability of confidence regions based on the likelihood ratio test statistic and for the coverage probability of highest posterior density regions. Conditions under which the two methods lead to identical regions, at least approximately, are also given.
Testing Exponentiality Based on Kullback-Leibler Information
Authors: Ebrahimi Nader , Habibullah Mohamed , Soofi Ehsan S.
Published: December 2018

In this paper a test of fit for exponentiality based on the estimated Kullback-Leibler information is proposed. The procedure is applicable when the exponential parameter is or is not specified under the null hypothesis. The test uses the Vasicek entropy estimate, so to compute it a ‘window size’ m must first be fixed. A procedure for choosing m for various sample sizes is proposed and corresponding critical values are computed by Monte Carlo simulations. The use of the proposed test is shown in an illustrative example. Also, by means of Monte Carlo simulations, the power of the proposed test under various alternatives is compared with that of other standard tests. The results are impressive and the proposed test, almost always, has higher power than that of the other tests considered.
Non-Homogeneous Autoregressive Processes for Tracking (Software) Reliability Growth, and Their Bayesian Analysis
Authors: Singpurwalla Nozer D. , Soyer Refik
Published: December 2018

We motivate a non-homogeneous autoregressive process as a model for reliability growth and consider two formulations, both Bayesian, which alleviate a limitation that least squares estimators for such processes of order greater than 1 exist only when repeated measurements of the time series are available. In one formulation, the prior assumption involves exchangeability of coefficients, whereas in the other an autoregressive structure is imposed on them. Both formulations enable us to cast the resulting processes in state space form for which the Kalman filter algorithm can be used. The first formulation involves an adaptation of standard techniques whereas the second results in a new methodology for adaptive filtering which is facilitated by an approximation due to Lindley. The procedures are demonstrated via a consideration of some data on software failures.
On a Kernel Approach to a Screening Problem
Authors: Boys Richard J.
Published: December 2018

Screening methods are developed which raise the proportion of individuals possessing an attribute in the retained population to a predetermined level. A specification region for the screening variable is derived by using a kernel smoothing approach. The sensitivity of the procedure is assessed within a range of parametric models. Comparisons are also made with empirical and estimative solutions. The kernel method is illustrated by an example of screening within a medical context.
An Exact Cholesky Decomposition and the Generalized Inverse of the Variance–Covariance Matrix of the Multinomial Distribution, with Applications
Authors: Tanabe Kunio , Sagae Masahiko
Published: December 2018

A symbolic formula is given for the square-root-free Cholesky decomposition of the variance–covariance matrix of the multinomial distribution. The evaluation of the symbolic Cholesky factors requires much fewer arithmetic operations than does the general Cholesky algorithm. Since the symbolic formula is not affected by an ill-conditioned matrix, it is particularly useful when the elements of a probability vector are of quite different orders of magnitude. A simpler formula is obtained for Pederson's procedure of sampling from a multinomial population. An explicit formula of the Moore–Penrose inverse of the variance–covariance matrix is given as well as a symmetric representation of a multinomial density approximation to the multinomial distribution. These formulae facilitate symmetric manipulation of the matrix and are useful in statistical modelling and computation involving the logistic density transformation of the multinomial distribution and in computer simulations of dynamic models in population genetics. Each element of the Cholesky factors is given interesting probabilistic interpretations.
Local Lyapunov Exponents: Looking Closely at Chaos
Authors: Wolff Rodney C. L.
Published: December 2018

When a deterministic mechanism gives rise to an erratic time series, then under certain conditions the series is said to exhibit chaos. The catalogue of statistical methods for the analysis of time series is being augmented currently by methods for the detection and quantification of deterministic chaos. A hallmark of chaos is the tendency of nearby trajectories to diverge in the short term. A measure of the average rate of exponential divergence exhibited by a chaotic system is given by the Lyapunov exponents of that system; the values of such exponents can suggest the presence of chaos. In this paper, we give a brief exposition of a finite time version of the Lyapunov exponent which is state dependent, and to which we attach the epithet ‘local'. The local Lyapunov exponent is shown to be able to detect non-linearity in some cases, and to have other qualitative features. As defined, its success is limited, but this does not preclude its usefulness as a diagnostic tool.
Chance and Chaos in Measles Dynamics
Authors: Grenfell B. T.
Published: December 2018

This paper examines the impact of seasonality and chaotic dynamics in simple models for the population dynamics of measles on the probability of fade-out of infection. It uses Monte Carlo simulations of the seasonally forced SEIR model, with parameters appropriate to a city of 1 million people. The incidence of fade-out in a spatially homogeneous model is compared with simple spatial models involving various degrees of coupling between subareas. The results indicate a significant degree of fade-out of infection, which is not consistent with previously derived criteria for the persistence of measles. Lowering the degree of spatial coupling does not substantially reduce the extent of fade-out. A simple non-linear analysis of the simulated series is presented, and the epidemiological implications of these results are discussed.
Estimation of Parameters in Heteroscedastic Linear Models
Authors: Mak Tak K.
Published: December 2018

This paper deals with linear regression models with non-homogeneous error variances. The common situation where the error variance is a smooth function of the values of the regressor variables is considered. The error variance function is represented by a function of known form, but its expression involves the vector of unknown regression coefficients β and an additional vector parameter θ. Estimation of β by iterative weighted least squares (IWLS) therefore also requires updating the estimate of θ at each iteration. When the error variance function is misspecified, different methods of updating θ would yield IWLS estimators of varying degrees of efficiency. Under such situations, a method of updating θ is proposed, resulting in an iterative procedure for estimating β that is asymptotically optimal in the class of IWLS estimators. When the form of the error variance function is correctly specified, the estimation procedure proposed is asymptotically equivalent to weighted least squares with known optimal weights. A real example of its application is given and the extension to general non-linear models is briefly indicated.
Log-Linear Models for Survey Data with Non-Ignorable Non-Response
Authors: Chambers R. L. , Welsh A. H.
Published: December 2018

We demonstrate the feasibility of constructing, interpreting and fitting computable log-linear models to categorical survey data with arbitrary non-nested patterns of non-ignorable non-response. Under our approach, the non-response probability for each cell of the classification defined by the categories of interest is modelled separately from the classification probability itself, and we adopt a model formulation which allows the non-response model to depend on scores, discrete covariates, continuous covariates or a mixture of types of covariate. We obtain explicit expressions for the score and information functions generated by the observed data which allow us to compute approximate standard errors and test statistics based on these functions. Through illustrative examples we quantify the fact that inferences obtained from different non-ignorable non-response models can vary considerably. This lends support to the calls in the literature for caution in using such models.
Large Sample Theory of Spacings Statistics for Tests of Fit for the Composite Hypothesis
Authors: Wells Martin T. , Jammalamadaka S. Rao , Tiwari Ram C.
Published: December 2018

Let X  1,…, Xn be a sequence of independent and identically distributed random variables with an unknown underlying continuous cumulative distribution function F. Often we would like to test a null hypothesis concerning the goodness of fit of F to some distribution function which is fully specified or belongs to some parametric family, i.e. in some applications the null hypothesis is simple whereas in others it may be composite. In this paper we present the large sample theory of tests based on non-symmetric functions of sample spacings under composite null hypotheses as well as under contiguous alternatives. Goodness-of-fit tests which are optimal within this class are constructed.
On Edgeworth Expansion and Bootstrap Confidence Bands in Nonparametric Curve Estimation
Authors: Hall Peter
Published: December 2018

The notion of pivoting and techniques of Edgeworth expansion have become basic tools for using and understanding bootstrap methods in finite parameter problems. This paper shows that these ideas have important contributions to make to infinite parameter problems, e.g. in the construction of simultaneous confidence bands for nonparametric density estimators. We propose a bootstrap method for confidence band construction based on a pivotal function estimator and illustrate its use. We give an Edgeworth expansion argument which demonstrates the efficacy of the technique. The expansion is a power series in {(nh)-1(log h  -1)3}1/2. where n is the sample size and h denotes the bandwidth. This should be compared with a power series in n  -1/2 in more conventional, finite dimensional problems.
A Note on the Calculation of Adjusted Profile Likelihood
Authors: Cox D. R. , Reid N.
Published: December 2018

An adjustment to the profile likelihood examined by Cox and Reid required orthogonalization of the nuisance parameters with respect to the parameter of interest. An approximation to the adjusted profile likelihood not requiring explicit orthogonalization is derived and exemplified.
Modelling Variance Heterogeneity: Residual Maximum Likelihood and Diagnostics
Authors: Verbyla A. P.
Published: December 2018

The assumption of equal variance in the normal regression model is not always appropriate. to attempt to eliminate unequal variance a transformation is often used but if the transformation is not successful, or the variances are of intrinsic interest, it may be necessary to model the variances in some way. We consider the normal regression model when log-linear dependence of the variances on explanatory variables is suspected. Detection of the dependence, estimation and tests of homogeneity based on full and residual maximum likelihood are discussed as are regression diagnostic methods based on case deletion and log-likelihood displacement. Whereas the behaviour of full and residual maximum likelihood is similar under case deletion, changes in residual maximum likelihood estimates and log-likelihood displacements tend to be smaller than maximum likelihood.
Dynamic Hierarchical Models
Authors: Gamerman Dani , Migon Hélio S.
Published: December 2018

An analysis of a time series of cross-sectional data is considered under a Bayesian perspective. Information is modelled in terms of prior distributions and stratified parametric linear models developed by Lindley and Smith and dynamic linear models developed by Harrison and Stevens are merged into a general framework. This is shown to include many models proposed in econometrics and experimental design. Properties of the model are derived and shrinkage estimators reassessed. Evolution, smoothing and passage of data information through the levels of the hierarchy are discussed. Inference with an unknown scalar observation variance is drawn and an extension to the non-linear case is proposed.
On the Behaviour of Wald Statistics for the Disjunction of Two Regular Hypotheses
Authors: Glonek G. F. V.
Published: December 2018

Occasionally hypotheses arise which, although not themselves regular, have the logical structure of the disjunction of two, or in general several, regular hypotheses. An example of such a hypothesis is that of collapsibility in a three-way contingency table as defined by Whittemore. When such a hypothesis is to be tested, a Wald test appears, superficially, to offer a convenient solution. However, closer inspection reveals that the irregularity intrinsic to such problems precludes the regularity conditions required for the valid construction of a Wald test statistic. The purpose of this paper is to describe explicitly the nature of the irregularity and its effect on the asymptotic behaviour of the Wald test statistic.
Solving Non-Linear Estimation Equations
Authors: Mak Tak K.
Published: December 2018

In this paper we consider the numerical computation of a vector parameter estimate θ^ which is a root of a system of unbiased non-linear estimation equations. A sequence {θ  (r)} is constructed which converges with a probability approaching 1 to θ^ from any starting value θ  (0). Unlike Newton-type algorithms, the θ  (r) do not involve matrices of partial derivatives. Furthermore it can be shown that θ(r)−θ^ is of the order Op(n  – r/2) so that great accuracy is achieved in very few iterations. When the θ  (r) do not have closed forms, an approximation is suggested. An alternative and usually simpler way is also found for deriving analytically the expected value of the matrix of partial derivatives in the asymptotic covariance matrix of θ^. The efficiency of the proposed algorithm is demonstrated numerically and compared with that of the EM and the Newton–Raphson algorithms. Applications to some well-known statistical problems, including missing data analysis, variance component models, familial correlations and errors-in-variables models are also briefly discussed.
Adjusted Versions of Profile Likelihood and Directed Likelihood, and Extended Likelihood
Authors: Barndorff-Nielsen O. E.
Published: December 2018

Several adjusted and invariant versions of profile likelihood and of directed likelihood (short for signed log-likelihood ratio) are discussed. These adjusted versions, which fall into two groups, are closely related to modified profile likelihood and modified directed likelihood. The first group has the advantage of not requiring an explicit specification of an ancillary, but partly at a cost of a loss of asymptotic accuracy in comparison with the modified versions. The second group consists of some asymptotically sharpened versions of modified profile likelihood. Furthermore, a concept of extended likelihood is proposed, which associates a likelihood-like function to certain combinants. At the end, the relationship between adjusted profile likelihoods on the one hand and adjusted directed likelihoods on the other is discussed briefly, and in this context a close connection is established between modified profile likelihood and the extended likelihood determined by the modified directed likelihood.
Parameterization of Continuous Time Autoregressive Models for Irregularly Sampled Time Series Data
Authors: Belcher J. , Hampton J. S. , Wilson G. Tunnicliffe
Published: December 2018

An increasingly valuable tool for modelling irregularly sampled time series data is the continuous time autoregressive model. The natural parameters in this model are the coefficients of the linear stochastic differential equation describing the process which gives rise to the data. A transformation of these parameters is introduced, based on the Cayley–Hamilton transformation. The new parameter space is identical with that of discrete time autoregressive models. The model is also modified by the introduction of prescribed moving average terms. The resulting modelling improvements include rapid and reliable convergence of parameter estimates and the ability to select the model order by testing whether the highest order coefficient is 0. A geophysical and a medical application illustrate the detection of periodicities in data by using the spectrum of the fitted model.
Affine Invariant Multivariate Multisample Sign Tests
Authors: Hettmansperger Thomas P. , Oja Hannu
Published: December 2018

Invariant sign-type tests for the multivariate, multisample location models are developed. The definition of sign is based on the Oja criterion function. Emphasis is on the two-sample location model and the associated equivariant estimate of shift. Asymptotic theory is developed to provide large sample approximations for the null distribution of the test and to provide a framework for a discussion of efficiency. Tests are then developed for the one-way and two-way lay-outs.
On the Convergence of Monte Carlo Maximum Likelihood Calculations
Authors: Geyer Charles J.
Published: December 2018

Monte Carlo maximum likelihood for normalized families of distributions can be used for an extremely broad class of models. Given any family {hθ: θ ∈ θ} of non-negative integrable functions, maximum likelihood estimates in the family obtained by normalizing the functions to integrate to 1 can be approximated by Monte Carlo simulation, the only regularity conditions being a compactification of the parameter space such that the evaluation maps θ → hθ(x) remain continuous. Then with probability 1 the Monte Carlo approximant to the log-likelihood hypoconverges to the exact log-likelihood, its maximizer converges to the exact maximum likelihood estimate, approximations to profile likelihoods hypoconverge to the exact profile and level sets of the approximate likelihood (support regions) converge to the exact sets (in Painlevé-Kuratowski set convergence). The same results hold when there are missing data if a Wald-type integrability condition is satisfied. Asymptotic normality of the Monte Carlo error and convergence of the Monte Carlo approximation to the observed Fisher information are also shown.
On the Geometric Convergence of the Gibbs Sampler
Authors: Roberts Gareth O. , Polson Nicholas G.
Published: December 2018

The rate of convergence of the Gibbs sampler is discussed. The Gibbs sampler is a Monte Carlo simulation method with extensive application to computational issues in the Bayesian paradigm. Conditions for the geometric rate of convergence of the algorithm for discrete and continuous parameter spaces are derived, and an illustrative exponential family example is given.
Quantifying the Influence of Initial Values on Non-Linear Prediction
Authors: Yao Qiwei , Tong Howell
Published: December 2018

Motivated by the m-step-ahead prediction problem in non-linear time series, a brief sketch of stochastic chaotic systems is provided. The accuracy of the prediction depends on the initial value, which is a typical feature of non-linear but not necessarily chaotic models. However, if the model is chaotic, small noise can be amplified very quickly through time evolution at some initial values, thereby decreasing dramatically the reliability of the prediction. Further, if the model is chaotic, small shifts in some initial values can lead to considerable errors in prediction, which can be monitored by the newly defined Lyapunov-like indices. For the nonparametric predictor constructed by the locally linear regression method, the mean-squared error may be decomposed into two parts: the conditional variance and the divergence resulting from a small shift in initial values. The decomposition also holds for more general predictors. A consistent estimator of the Lyapunov-like index is also constructed by the locally linear regression method. Both simulated and real data are used as illustrations.
The Cholesky Decomposition of P – Pp′
Authors: Farebrother Richard William
Published: December 2018

In this paper we outline an alternative derivation of the square-root-free Cholesky decomposition of P – pp′ based on square-root-free variants of a standard algorithm in statistical computing.
Finding Words with Unexpected Frequencies in Deoxyribonucleic Acid Sequences
Authors: Prum Bernard , Rodolphe François , De Turckheim Elisabeth
Published: December 2018

Considering a Markov chain model for deoxyribonucleic acid sequences, this paper proposes two asymptotically normal statistics to test whether the frequency of a given word is concordant with the first-order Markov chain model or not. The problem is to choose estimates μ^(W) of the expectation of the frequency Mw of a word W in the observed sequence such that the asymptotic variance of MW−μ^(W) is easily computable. The first estimator is derived from the frequency of W  [– 1], which is W with its last letter deleted. The second, following an idea of Cowan, is the conditional expectation Mw given the observed frequencies of all two-letter words. Two examples on phage lambda and phage T7 are shown.
Fast Approximate Maximum A Posteriori Restoration of Multicolour Images
Authors: Ferrari Pablo A. , Frigessi Arnoldo , Sá Paula Gonzaga
Published: December 2018

We propose a new algorithm for the approximation of the maximum a posteriori (MAP) restoration of noisy images. The image restoration problem is considered in a Bayesian setting. We assume as prior distribution multicolour Markov random fields on a graph whose main restriction is the presence of only pairwise site interactions. The noise is modelled as a Bernoulli field. Computing the mode of the posterior distribution is NP complete, i.e. can (very likely) be done only in a time exponential in the number of sites of the underlying graph. Our algorithm runs in polynomial time and is based on the coding of the colours. It produces an image with the following property: either a pixel is coloured with one of the possible colours or it is left blank. In the first case we prove that this is the colour of the site in the exact MAP restoration. The quality of the approximation is then measured by the number of sites being left blank. We assess the performance of the new algorithm by numerical experiments on the simple three-colour Potts model. More rigorously, we present a probabilistic analysis of the algorithm. The results indicate that the approximation is quite often sufficiently good for the interpretation of the image.
Approximate Random Variate Generation from Infinitely Divisible Distributions with Applications to Bayesian Inference
Authors: Damien Paul , Laud Purushottam W. , Smith Adrian F. M.
Published: December 2018

Stochastic processes with independent increments play a central role in Bayesian nonparametric inference. The distributions of the increments of these processes, aside from fixed points of discontinuity, are infinitely divisible and their Laplace and/or Fourier transforms in the Lévy representation are usually known. Conventional Bayesian inference in this context has been limited largely to providing point estimates of the random quantities of interest, although Markov chain Monte Carlo methods have been used to obtain a fuller analysis in the context of Dirichlet process priors. In this paper, we propose and implement a general method for simulating infinitely divisible random variates when their Fourier or Laplace transforms are available in the Lévy representation. Theoretical justification is established by proving a convergence theorem that is a ‘sampling form’ of a classical theorem in probability. The results provide a method for implementing Bayesian nonparametric inference by using a wide range of stochastic processes as priors.