Showing papers in "Journal of the royal statistical society series b-methodological in 1968"

A generalization of bayesian inference

Abstract: Procedures of statistical inference are described which generalize Bayesian inference in specific ways Probability is used in such a way that in general only bounds may be placed on the probabilities of given events, and probability systems of this kind are suggested both for sample information and for prior information These systems are then combined using a specified rule Illustrations are given for inferences about trinomial probabilities, and for inferences about a monotone sequence of binomial pi Finally, some comments are made on the general class of models which produce upper and lower probabilities, and on the specific models which underlie the suggested inference procedures

A General Definition of Residuals

Abstract: RESIDUALS are now widely used to assess the adequacy of linear models; see Anscombe (1961) for a systematic discussion of significance tests based on residuals, and for references to earlier work. A second and closely related application of residuals is in time-series analysis, for example in examining the fit of an autoregressive model. In the context of normal-theory linear models, the n x 1 vector of random variables Y is assumed to have the form

A Simplified Monte Carlo Significance Test Procedure

Abstract: JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Wiley and Royal Statistical Society are collaborating with JSTOR to digitize, preserve and extend access to Journal of the Royal Statistical Society. Series B (Methodological). SUMMARY The use of Monte Carlo test procedures for significance testing, with smaller reference sets than are now generally used, is advocated. It is shown that, for given oX = l/n, n a positive integer, the power of the Monte Carlo test procedure is a monotone increasing function of the size of the reference set, the limit of which is the power of the corresponding uniformly most powerful test. The power functions and efficiency of the Monte Carlo test to the uniformly most powerful test are discussed in detail for the case where the test criterion is N(y, 1). The cases when the test criterion is Student's t-statistic and when the test statistic is exponentially distributed are considered also.

The Ergodic Theory of Subadditive Stochastic Processes

Abstract: SUMMARY An ergodic theory is developed for the subadditive processes introduced by Hammersley and Welsh (1965) in their study of percolation theory. This is a complete generalization of the classical law of large numbers for stationary sequences. 1. SUBADDITIVE PROCESSES IN an important paper Hammersley and Welsh (1965) introduced the concept of a subadditive stochastic process, and they have shown how such processes arise naturally in various contexts, but particularly in the study of random flows in lattices. They have shown that one may expect these processes to exhibit a certain ergodic behaviour, and have taken the first steps towards the construction of an ergodic theory like the classical one for averages of stationary sequences. If T is any subset of the real line, a subadditive process x on T is a collection of (real) random variables xst(s, t E T, s < t) with the property that

The Choice of Variables in Multiple Regression

Abstract: Professor R. L. PLACKETT in the Chair] SUMMARY This paper is concerned with the analysis of data from a multiple regression of a single variable, y, on a set of independent variables, xl, x2, .. . ,xr. It is argued that the form of the analysis should depend on the use that is to be made of the regression, and that therefore an approach employing ideas from decision theory may be worth while. Two situations are analysed in this way: in the first it is desired to predict a future value of y; in the second we wish to control y at a preassigned value. The two analyses are found to be different: in particular, the standard errors of the regression coefficients are found to be irrelevant in the prediction problem, but not in the control problem. In the former it is shown that, under rather special assumptions on the multiple regression experiment, the analysis is similar to that recommended by other writers. If the costs of control do not depend on the values at which the control takes place, a similar analysis holds for the second problem. The approach throughout is Bayesian: there is no discussion of this point, I merely ask the non-Bayesian reader to examine the results and consider whether they provide sensible and practical answers.

Missing Data in Regression Analysis

Abstract: SUMMARY Two alternative methods for dealing with the problem of missing observations in regression analysis are investigated. One is to discard all incomplete observations and to apply the ordinary least-squares technique only to the complete observations. The alternative is to compute the covariances between all pairs of variables, each time using only the observations having values of both variables, and to apply these covariances in constructing the system of normal equations. The former is shown to be equivalent to the Fisher-Yates method of assigning "neutral" values to missing entries in experimental design. The investigation is carried out by means of simulation. Eight sets of regression data were generated, differing from each other with respect to important factors. Various deletion patterns are applied to these regression data. The estimates resulting from applying the two alternative methods to the data with missing entries are compared with the known regression equations. In almost all the cases which were investigated the former method (ordinary least squares applied only to the complete observations) is judged superior. However, when the proportion of incomplete observations is high or when the pattern of the missing entries is highly non-random, it seems plausible that one of the many methods of assigning values to the missing entries should be applied.

Models for the Response of a Mixture

Abstract: SUMMARY Care needs to be exercised in the choice of model for a mixture system. The polynomial model, for example, cannot satisfactorily account for components which are inert or have additive effects and its coefficients lose their interpretative value when the variables are the proportions of components in the mixture. This paper proposes models constructed from functions homogeneous of degree one which overcome these difficulties, yet leave a wide choice of models.

The Combination of Information in Generally Balanced Designs

Abstract: A general method is proposed for combining information in generally balanced designs when estimates of treatment effects are available from more than one stratum. Estimates of stratum variances are obtained by equating residual sum of squares in each stratum to their expectations, using the ML estimates of treatment effects. Weights may be given, the solution then being explicit, or fitted, in which case iteration is required. The results of a Monte Carlo study are given for a BIB design to test the distribution of the variance estimates.

A Note on the Efficiency of Least-squares Estimates

Abstract: SUMMARY A linear model is considered in which errors are independent and identically distributed with zero mean. If the error distribution is specified, except possibly for unknown parameters, the asymptotic efficiency of least-squares estimates relative to maximum-likelihood estimates can be found. For regression parameters orthogonal to the general mean the asymptotic efficiency, which is independent of the design matrix, is calculated explicitly for an Edgeworth series, for a Pearson Type VII distribution and for a log gamma distribution.

Experiments with Mixtures: A Generalization of the Simplex‐Lattice Design

Abstract: The purpose of these designs is the empirical prediction of the response to any mixture of the components when the response depends only on the proportions of the components present, but not on the total amount of the mixture. Gorman and Hinman (1962) applied and extended the simplex-lattice design. There may be problems in practice in which the n components with proportions (1.1) are mixtures of several other components with proportions

A Relatively Simple Form of the Distribution of the Multiple Correlation Coefficient

Abstract: SUMMARY A series is obtained for the cumulative distribution function of the multiple correlation coefficient in which the coefficients are interpretable as binomial probabilities. When N-p is even (N is the sample size and p is the number of components in the underlying multivariate normal distribution) the series is finite with ji(n -p) + 1 terms. Other series are also considered. A comparatively simple approximation to the general distribution of the multiple correlation coefficient is presented which, in particular for p = 2, is more accurate than Fisher's z-transform over a large set of values of the parameters involved.

On a Distribution associated with Certain Stochastic Processes

Abstract: SUMMARY The distribution discussed is (a) the total number of games lost by the ruined gambler starting with a monetary units against an infinitely rich adversary in the classical ruin problem; (b) the total size of an epidemic in an infinite population starting with a cases and with constant probabilities of infection and recovery; (c) the number of customers served in a busy period (starting with a customers) of the queue M/M/1. These derivations, the relationships between them and various properties of the distribution are discussed. HAIGHT (1961a) examined the distribution, analogous to the Borel-Tanner, of the number of members of the queue M/M/1 served during the busy period starting with a members. We show that it is the distribution of the total number of steps taken in the direction of the origin (i.e. the number of games lost by the ruined gambler) in the classical ruin random walk and that it is also McKendrick's (1926) distribution of the total size of an epidemic in an infinite population with constant probabilities of success and failure (and a initial cases). After considering each of these models and showing that both the other two may be transformed into the ruin random walk, we derive some properties of the distribution.

Rank-order Tests of Linear Hypotheses

Abstract: SUMMARY This paper considers tests for linear hypotheses using the ranks of independent observations from an unspecified distribution. By means of results on non-linear estimation of the regression vector P it is shown that the decomposition of, and tests for the sum of squares of ranks proceeds analogously to the usual analysis of variance model. These results are then applied to the Wilcoxon and Kruskal-Wallis tests and a multiple rank correlation coefficient.

Confidence Properties of Bayesian Interval Estimates

Abstract: SUMMARY Coverage probabilities, in the confidence theory sense, of Bayes interval estimates constructed according to different criteria are examined. Suggestions are made for the choice of a prior density in the case of complete ignorance concerning the parameter under estimation. These densities are members of the family of relatively invariant densities discussed by Hartigan (1964). Confidence properties of a related family are also discussed.