Showing papers in "Journal of the royal statistical society series b-methodological in 1973"

Combining Possibly Related Estimation Problems

Abstract: SUMMARY We have two sets of parameters we wish to estimate, and wonder whether the James-Stein estimator should be applied separately to the two sets or once to the combined problem. We show that there is a class of compromise estimators, Bayesian in nature, which will usually be preferred to either alternative. "The difficulty here is to know what problems are to be combined togetherwhy should not all our estimation problems be lumped together into one grand melee ?" GEORGE BARNARD commenting on the James-Stein estimator, 1962.

Testing Transformations to Normality

Abstract: SUMMARY Andrews's exact test for the value of the parameter in the Box and Cox parameteric family of transformations is compared with two tests derived from the likelihood function. These two tests are shown, for a numerical example, to be uniformly more powerful than the exact test.

Central Limit Analogues for Markov Population Processes

Abstract: 1 SUMMARY In this paper the main discussion is concerned with obtaining asymptotic results for sequences of birth and death processes which are similar to the central limit theorem for sequences of univariate random variables The motivation is the need to obtain useful approximations to the distributions of sample paths of processes which arise as models for population growth, but for which Kolmogorov differential equations are intractable In the first section, univariate processes are considered, and conditions are given for the weak convergence of Z N (t) = {X N (t) - aN}/N, where {X N (t), N = 1,2,…} is a sequence of ergodic birth and death processes, to those of an Ornstein-Uhlenbeck process N → ∞ A heuristic method is given which may help explain why this convergence holds, and some examples are given for purposes of illustration The second part deals with multivariate processes, and three examples are considered in detail: a model for the growth of the sexes in a biological population, a multivariate Ehrenfest process, and a model for the growth and interreaction of two cities The paper concludes with a discussion of various related results It is shown that in certain special cases it is possible to obtain diffusions other than the Ornstein-Uhlenbeck process as limits Finally, heavy traffic results are included for congestion situations originally considered in the special case of time-homogenous arrival rates by Kingman Transient processes such as epidemics are also shown to exhibit a “central limit” behavior

Bayesian and Classical Analysis of Poisson Regression

Abstract: SUMMARY This paper is concerned with the problem of testing the existence of a trend in the means Gi of Poisson distributions. It is assumed that these means are changing exponentially, that is, log Gi = ci+/x2. A classical method is reviewed which is used for testing the hypothesis P = 0. The exact Bayesian distribution for P is derived and a Bayesian approximation suggested which proved to be very useful. Finally, a comparison of these three methods by means of numerical examples is made.

Simultaneous Confidence Intervals for Product‐Type Interaction Contrasts

Abstract: SUMMARY The set of product-type interaction contrasts, which contains the subsets of interaction residuals, tetrad contrasts, double-dichotomy contrasts and pooled-tetrad contrasts, is discussed. In the Normal case with equal and known variances, simultaneous confidence intervals for a given set of such contrasts can be constructed in several ways. Five such methods (the Scheffe method, the Tukey method, the Dunn method, a method based on Roy's maximum-root statistic and a modification of Tukey's method) are presented and compared in terms of the widths of the resulting 95 and 99 per cent confidence intervals. The Dunn method is found to yield the shortest intervals for interaction residuals, for tetrad contrasts and for doubledichotomy contrasts. On the other hand, for the set of pooled-tetrad contrasts, and therefore also for any larger set of product-type interaction contrasts, the maximum-root method yields the shortest intervals. A table of the half-widths of the intervals given by the recommended methods is provided.

Bayesian Spline Regression When the Number of Knots is Unknown

Abstract: SUMMARY The problem of selecting the model and coefficients for a spline regression curve with normal errors is considered. The locations of all possible knots are assumed to be known; which subset of these is the set of actual knots is unknown. Each subset forms a model. Priors of the form of a marginal distribution on the index of the model and a conditional distribution for each model on the coefficients and variance are allowed. The posterior distribution is explicitly derived for natural conjugate and vague priors. The optimal predictor is derived for a loss function which is a generalization of that appearing in Lindley (1968). All that is required of the opinion is the marginal distribution for the model and the mean vector for the coefficients.

Survey Design, Symmetry and Posterior Distributions

Abstract: SUMMARY It is now well known that the sample design is irrelevant in a Bayesian analysis of survey data provided the units are labelled and the sampling is non-informative. However, information about the labels is often not available for the analysis, and in this situation the design will contain information about the labelling and hence affect the inferences. There is one important exception: if the prior distribution is exchangeable and we are interested only in symmetric functions of the unit values then the design is still irrelevant even without the labels.