Journal of the Royal Statistical Society Series B (Statistical Methodology)

Atkinson, A. C.; Kent, J. T.

doi: 10.1111/j.2517-6161.1993.tb01465.xpmid: N/A

Smith, A. F. M.; Roberts, G. O.

doi: 10.1111/j.2517-6161.1993.tb01466.xpmid: N/A

The use of the Gibbs sampler for Bayesian computation is reviewed and illustrated in the context of some canonical examples. Other Markov chain Monte Carlo simulation methods are also briefly described, and comments are made on the advantages of sample‐based approaches for Bayesian inference summaries.

Besag, Julian; Green, Peter J.

doi: 10.1111/j.2517-6161.1993.tb01467.xpmid: N/A

Markov chain Monte Carlo (MCMC) algorithms, such as the Gibbs sampler, have provided a Bayesian inference machine in image analysis and in other areas of spatial statistics for several years, founded on the pioneering ideas of Ulf Grenander. More recently, the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a Markov random field has opened the way to the use of MCMC in general Bayesian computation. In this paper, we trace the early development of MCMC in Bayesian inference, review some recent computational progress in statistical physics, based on the introduction of auxiliary variables, and discuss its current and future relevance in Bayesian applications. We briefly describe a simple MCMC implementation for the Bayesian analysis of agricultural field experiments, with which we have some practical experience.

Gilks, W. R.; Clayton, D. G.; Spiegelhalter, D. J.; Best, N. G.; McNeil, A. J.; Sharples, L. D.; Kirby, A. J.

doi: 10.1111/j.2517-6161.1993.tb01468.xpmid: N/A

We review applications of Gibbs sampling in medicine, involving longitudinal, spatial, covariate measurement and survival models. Applications in immunology, pharmacology, transplantation, cancer screening, industrial epidemiology and genetic epidemiology are discussed.

doi: 10.1111/j.2517-6161.1993.tb01469.xpmid: N/A

Grunwald, Gary K.; Raftery, Adrian E.; Guttorp, Peter

doi: 10.1111/j.2517-6161.1993.tb01470.xpmid: N/A

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.

Fisher, N. I.; Lunn, A. D.; Davies, S. J.

doi: 10.1111/j.2517-6161.1993.tb01471.xpmid: N/A

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.

Macaskill, G. T.

doi: 10.1111/j.2517-6161.1993.tb01472.xpmid: N/A

The problem of partial non‐linear regression is used to provide an example of McCullagh and Tibshirani's profile likelihood adjustment for more than one parameter of interest, where for one linear nuisance parameter an expression for the adjusted profile likelihood can be derived.

McCabe, B. P. M.; Leybourne, S. J.

doi: 10.1111/j.2517-6161.1993.tb01473.xpmid: N/A

This paper addresses the problem of testing for purely random parameter variation in nonlinear regression models. Based on different approximations to the true density of the data, score‐type tests are constructed and their asymptotic distributions are derived. The local power of the tests is investigated both theoretically and via Monte Carlo simulation. An empirical testing example, involving a well‐known non‐linear aggregate demand for money function, is also given.

Eubank, R. L.; Thomas, Will

doi: 10.1111/j.2517-6161.1993.tb01474.xpmid: N/A

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.