Showing papers in "Journal of the royal statistical society series b-methodological in 1985"

Some Aspects of the Spline Smoothing Approach to Non-Parametric Regression Curve Fitting

Abstract: Non-parametric regression using cubic splines is an attractive, flexible and widely-applicable approach to curve estimation. Although the basic idea was formulated many years ago, the method is not as widely known or adopted as perhaps it should be. The topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology. Particular subjects covered include the basic principles of the method; the relation with moving average and other smoothing methods; the automatic choice of the amount of smoothing; and the use of residuals for diagnostic checking and model adaptation. The question of providing inference regions for curves-and for relevant properties of curves--is approached via a finite-dimensional Bayesian formulation.

Kernels for Nonparametric Curve Estimation

Abstract: SUMMARY The choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated Explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error, IMSE (the present proof of the optimality of the latter kernels is restricted up to order k = 5) These kernels are also of interest for the nonparametric estimation of probability densities and spectral densities A finite sample study indicates that higher order kernels-asymptotically improving the rate of convergence-may become attractive for realistic finite sample size Suitably modified kernels are considered for estimating at the extremities of the data, in a way which allows to retain the order of the bias found for interior points

A model for high-order Markov chains

Abstract: On introduit un modele pour des chaines de Markov du leme ordre qui comporte seulement un parametre additionnel pour chaque retard

Stochastic Models of Computer Communication Systems

Abstract: (Read before the Royal Statistical Society at a meeting organized by the Research Section on Wednesday, May 8th, 1985, Professor J. B. Copas in the Chair) SUMMARY This paper describes some examples of the stochastic models found useful in the design and analysis of advanced computer and communication systems. Our major theme might be termed the control of contention. As illustrations of this theme we discuss concurrency control procedures for databases, dynamic channel assignment for cellular radio, and random access schemes for the control of a broadcast channel. We emphasize asymptotic properties of product-form distributions and we present some new results on the stability of acknowledgement based random access schemes. This paper is intended to describe to the Society some examples of the stochastic models found useful in the design and analysis of advanced computer and communication systems. The examples chosen are broadly concerned with what might be termed the control of contention, and an attempt has been made to provide enough of the technical background to motivate the models considered. In Section 2 we describe a probabilistic model, due to Mitra (1985), for conflicts anlong tran- sactions in a database. Such conflicts can arise in distributed computer systems, where to ensure the consistency of a database it is often necessary to forbid the concurrent execution of transactions involving common items: a transaction must then contend with other transactions for access to the items it requires. Mitra (1985) has shown that his model can be used to answer some important design questions concerning concurrency control procedures which use exclusive and non-exclusive locks. Mitra's results are based upon a product-form solution; we indicate how his asymptotic formulae can be extended beyond the range of light traffic and the assumption of an unstructured database. In Section 3 we discuss one of the many interesting problems which arise in connection with cellular radio. Cellular radio makes efficient use of a limited number of radio channels by allowing the repeated reuse of each channel in sufficiently separated spatial regions. The topic we consider is contention between different regions for the use of dynamically assigned channels. Everitt and Macfadyen (1983) have described an analytically tractable method of dynamic channel assign- ment, which they term the maximum packing strategy. Again a product form solution is involved: from this it is easy to obtain asymptotic formulae applicable in light traffic. These formulae establish the advantage of the strategy over a fixed channel assignment for low enough loss pro- babilities, but the advantage disappears as traffic and the number of channels increase. The real potential of dynamic schemes is their ability to cope automatically with traffic intensities which fluctuate in space and time.

Estimation and Hypothesis Testing in Finite Mixture Models

Abstract: SUMMARY Finite mixture models are a useful class of models for application to data. When sample sizes are not large and the number of underlying densities is in question, likelihood ratio tests based on joint maximum likelihood estimation of the mixing parameter, X, and the parameter of the underlying densities, 0, are problematical. Our approach places a prior distribution on X and estimates 0 by maximizing the likelihood of the data given 0 with X integrated out. Advantages of this approach, computational issues using the EM algorithm and directions for further work are discussed. The technique is applied to two examples.

Variance Component Models with Binary Response: Interviewer Variability

Abstract: SUMMARY Interviewer variability in a binary response is an example of a problem requiring variance component estimation in a non-normal family. The maximum likelihood estimation procedure is derived and used to examine some binary items on a large questionnaire. This raises some interesting questions about the use of unbalanced ANO VA methods with these data. Two interviewers asking the same question of the same respondent can obtain different responses. Such interviewer effects create problems in the design and analysis of survey questionnaires. Collins (1980) gives a review of these problems together with a variety of partial remedies. These include pilot studies and careful design to eliminate difficulties with wording of questions and instructions on the questionnaires. In spite of these precautions, interviewer effects remain. Questionnaire items are usually categorical, and occasionally binary, but usually for analysis the questionnaire item is reduced to a binary response. It is the variability of the population of interviewers that is of interest and not just the individual interviewers, thus this variability is best modelled as a random effect rather than a fixed effect. The number of interviews completed by each interviewer varies, so that the design is unbalanced. Thus we have an unbalanced variance component model with a categorical or binary response. The intraclass correlation-the ratio of the interviewer variance to the sum of all variances in the model-is of great interest as a measure of interviewer variability. In estimating the variance components, the categorical or binary nature of the response is usually ignored, and the analysis carried out using analysis of variance. This is not maximum likelihood, but analysis of variance does have some desirable properties when the error variables are non-normal. The rule-of-thumb generally applied is that the analysis of variance is reasonably accurate as long as the proportions in each of the categories of the response are between 0.1 and 0.9. Scheffe (1959, p. 343) discusses the effects of non-normality in a one-way balanced model. Very little theoretical work has been

Analysis of Field Experiments by Least Squares Smoothing

Abstract: SUMMARY Assuming a smooth trend plus independent error model for the environmental effects in the yields of a field plot experiment, least squares smoothing methods are developed to estimate both the treatment effects and the unknown trend Treatment estimates are closely related to those resulting from a generalized least squares analysis in which the covariance structure for the environmental effects has a particular form However, the main emphases are on the accuracy of treatment estimates under a fixed smooth trend plus error model and the exploratory power of the basic method to isolate trend effects of unknown form Although the detailed development is for the one-dimensional case, generalizations of the smoothness concept and extensions to two dimensions are also discussed Application of the basic method is illustrated on three data sets and the results compared with other analyses

On Asymptotic Normality of Limiting Density Functions with Bayesian Implications

Abstract: Walker (1969) presented a straightforward approach to asymptotic normality of posterior distributions based on independent and identically distributed (i.i.d.) observations. Dawid (1970) extended this work by providing a weaker set of conditions to cover some distributions whose ranges depend on the parameters of interest. More recently, Heyde and Johnstone (1979) simplified Walker's conditions for stochastic processes. In this paper, we introduce three basic normality conditions for limiting density functions. In the Bayesian context, our approach is data-determined and is nonprobabilistic in nature. Nevertheless, the results thus obtained can be easily converted to probabilistic ones when compared with those given by other investigators. It is shown that our conditions as a whole have a greater generality and flexibility than those previously assumed either by Dawid or Heyde and Johnstone. In particular, for the i.i.d. case, the commonly assumed Wald-type uniform-boundedness condition for tail behaviours is shown to be unnecessary for asymptotic normality (Section 3.3). The advantage of our approach is further exemplified by a simple proof of the asymptotic normality of distributions conjugate to the exponential family.

Locally Best Invariant Tests of the Error Covariance Matrix of the Linear Regression Model

Abstract: SUMMARY This paper considers a class of hypothesis testing problems concerning the covariance matrix of the disturbances in the classical linear regression model. A test that is locally best invariant against one-sided alternative hypotheses is constructed and shown to be identical to a one-sided version of the Lagrange Multiplier test.

Modelling and Residual Analysis of Nonlinear Autoregressive Time Series in Exponential Variables

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

A Note on Restricted Maximum Likelihood Estimation with an Alternative Outlier Model

Abstract: SUMMARY For single outliers in normal theory fixed effects models a mean slippage model is commonly used. An alternative is to model the outlier as arising from an unknown observation with inflated variance. Maximum likelihood estimates for the position of the outlier under the two models need not agree. This paper considers maximizing a restricted part of the likelihood to estimate the variance parameters and character- izes these estimates in terms of standard least squares parameters. It is shown that the residual variance and outlier position are the same under both models. In a recent paper Cook, Holschuh and Weisberg (1982) consider two models for single outliers in fLxed effects linear models. One is based on the assumption that contamination gives rise to slippage in the expected values of the observations (Weisberg, 1980, Section 5.3). Cook et al. point out the key role of Studentized residuals in this model, for instance in estimating the position of an outlier or testing for the presence of an outlier. Alternatively one can assume that an outlier arises from an error term with an increased variance. Cook et al. argue intuitively that it seems reasonable that Studentized residuals might play a similar role under this alternative model but show that maximum likelihood estimates of outlier position can differ under these two models. Cook et al. note that their models can be fitted into a linear model framework discussed by Harville (1977). They do not note that Harville recommends a restricted maximum likelihood (REML) approach using only a restricted part of the likelihood to estimate the variance para- meters. Patterson and Thompson (1971) noted that this REML approach takes account of loss of degrees of freedom in estimating flxed effects. In this note, REML estimates for the alternative model are derived and expressed in standard least squares statistics. It is shown that the observation having the largest Studentized residual is the one picked out as the outlier.

Recursive Estimation of Dynamic Linear Models

Abstract: SUMMARY A dynamic, linear model for the-analysis of univariate time series is proposed. It encompasses many of the common statistical models as special cases such as multiple regression, exponential smoothing and mixed autoregressive-moving average processes. Its distinguishing feature is that it relies on only one primary source of randomness. It therefore not only provides a simpler framework for the study of dynamic models but also eliminates the need for the contentious system variance matrix which has been credited with hampering the use of recursive forecasting methods in practice. The associated Kalman filter is also derived. This paper considers the issue of the recursive estimation of a dynamic linear statistical model (DLM). Its primary antecedent is the work of Duncan and Horn (1972) which in turn is based extensively on the pioneering ideas of Kalman (1960). The scope of the paper is restricted to time invariant dynamic linear models with only one observable dependent variable. Recently, Ameen and Harrison (1983) have outlined difficulties associated with Kalman filter- ing which have inhibited its use in the context of Bayesian forecasting. The "specification of the associated system variance matrices has proven a major obstacle" and they cite reasons such as its non-uniqueness together with the fact that it is not invariant to scale changes of the independent variables. This paper outlines an approach which eliminates the need for the system variance matrix, relying instead on what is best called a "permanent effects" vector whose ele- ments resemble the role of smoothing constants in exponential smoothing. Throughout the paper lower case letters are used to represent scalars and vectors, capital letters are used for matrices, and Greek letters are reserved for parameters. All vectors are of the column variety and their transpose is indicated in the usual way with a prime. The letter E denotes the expectation operator, I represents an identity matrix and carets are used to indicate estimators.

Gaussian Estimation for Correlated Binomial Data

Abstract: SUMMARY The mean/variance structure of correlated binomial data is discussed, and Whittle's (1961) Gaussian estimation is suggested as a useful general method in this context. Comparisons are made with other monrent methods in current use.

A Distributional Model for Repeated Failure Time Measurements

Abstract: SUMMARY A standard model for repeated measurements is based on the multivariate Normal distribution with equicorrelated variables. A corresponding analysis is developed for Weibull failure times. The resulting distribution has some attractive properties for practical application to multivariate failure times.

Diagnostics for Smoothing Splines

Abstract: SUMMARY Diagnostic measures appropriate for use with smoothing splines are derived and their properties are investigated. The proposed measures focus on detection of observations which substantially influence the fit and provide additional information over that obtained from examination of residuals alone. A numerical example illustrates the technique.

Maximum Likelihood Estimation of the Lower Tail of a Probability Distribution

Abstract: SUMMARY We consider the estimation of the three parameters of the lower tail of a distribution function based on the k smallest out of n observations. The likelihood function has a singularity but it is argued that a local maximum, when it exists, should be taken as the m.l.e. Asymptotic results as k - oo, n -+ oo, k/n -+ 0 show that such a local maximum does exist, and provides consistent estimators whenever the shape parameter is greater than one; otherwise there is no local maximum and likelihood inference fails. We also discuss interval estimation and propose a test to distinguish between the Type I and Weibull limit laws.

A General Class of Distributions on the Simplex

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

On the Estimation of Residual Variance and Order in Autoregressive Time Series

Abstract: SUMMARY We study the bias of Yule-Walker, least squares and Burg-type estimates of the residual variance of autoregressive processes. Both simulations and theory indicate that YuleWalker estimates are inferior to least squares and Burg-type estimates. The effect on order determination is also studied, and we extend the results on overestimation of the AIC criterion to the general multivariate case. For strongly autocorrelated processes, Yule-Walker estimates of residual variance and order may be severely biased even for comparatively large sample sizes.

Bounds on Reciprocal Moments with Applications and Developments in Stein Estimation and Post‐Stratification

Abstract: SUMMARY Bounds on the kth reciprocal moment of a distribution are given in terms of the mean, variance and left extremity of the distribution. A methodology for obtaining distribution-specific bounds, with improved accuracy, is outlined. Some natural approximations are noted. Applications are given in Stein estimation, and in classical and Bayesian post-stratification.

Perturbation Theory for Generalized Procrustes Analysis

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

Asymptotic Expansions for Confidence Limits in the Presence of Nuisance Parameters, with Applications

Abstract: SUMMARY Series expansions for confidence limits based on consistent estimators are obtained in the presence of nuisance parameters. The maximum likelihood estimator is treated as a special case and the necessary characteristics of its sampling distribution are obtained in terms of basic quantities derivable from the likelihood function. The theory is applied to the problem of setting separate confidence limits for the scale and shape parameters of the Weibull distribution. Comparisons are made with earlier simulation studies.

A criterion for the construction of optimal neighbour designs

Abstract: On donne une methode a deux etapes ou le plan de blocs incomplets resolvable efficace peut etre utilise comme point de depart pour la construction d'un plan de voisinage efficace

Modelling Data Containing Exact Zeroes Using Zero Degrees of Freedom

Abstract: On etablit l'existence et l'unicite des estimations du maximum de vraisemblance pour la distribution d'echelle du khi-deux non centre a zero degre de liberte

On the Polynomial Functional Relationship

Abstract: SUMMARY A method for obtaining consistent estimators of the coefficients in a polynomial functional relationship model with normal errors is proposed when the error covariance matrix is either completely known or known up to a proportionality factor. For quadratic functional relationships with possibly non-normal errors, we present a consistent estimator when the error covariance matrix is known. A large sample test of the quadratic against the linear functional relationship is also suggested in this case.

Bayesian Estimation of Finite Population Parameters in Categorical Data Models Incorporating Order Restrictions

Abstract: This note describes a Bayesian method for estimation of finite population parameters in general population surveys where acceptable regression-type models are typically unavailable. A categorical data model is adopted as in Ericson (1969, Section 4). However, specifications of smoothness are incorporated into the prior distribution. These smoothness conditions are expressed as unimodal or, possibly, multi-modal order relations among the category probabilities. Emphasis is placed on posterior inference about the finite population mean. Of independent interest is the methodology for evaluating the posterior moments and probabilities using Monte Carlo integration with importance sampling.