Consider a linear regression model with independent normally distributed errors. Suppose that the scalar parameter of interest is a specified linear combination of the components of the regression parameter vector. Also suppose that we have uncertain prior information that a distinct specified linear combination of these components takes the value zero. We provide succinct and informative descriptions of interval estimators for the parameter of interest using the new concepts of *scaled offset* and *scaled half-length*. We describe the Bayesian equi-tailed and shortest credible intervals for the parameter of interest that result from a prior density for the parameter about which we have uncertain prior information that is a mixture of a rectangular ‘slab’ and a Dirac delta function ‘spike’, combined with noninformative prior densities for the other parameters of the model. This prior belongs to the class of ‘slab and spike’ priors, which have been used for Bayesian variable selection. We compare these credible intervals with Kabaila and Giri's frequentist confidence interval for the parameter of interest that utilizes this uncertain prior information. We show that these frequentist and Bayesian interval estimators depend on the data in very different ways. We also consider some close variants of this prior distribution that lead to Bayesian and frequentist interval estimators with greater similarity. Nonetheless, as we show, substantial differences between these interval estimators remain.

The Fay–Herriot model is a standard model for direct survey estimators in which the true quantity of interest, the superpopulation mean, is latent and its estimation is improved through the use of auxiliary covariates. In the context of small area estimation, these estimates can be further improved by borrowing strength across spatial regions or by considering multiple outcomes simultaneously. We provide here two formulations to perform small area estimation with Fay–Herriot models that include both multivariate outcomes and latent spatial dependence. We consider two model formulations. In one of these formulations the outcome-by-space dependence structure is separable. The other accounts for the cross dependence through the use of a generalized multivariate conditional autoregressive (GMCAR) structure. The GMCAR model is shown, in a state-level example, to produce smaller mean square prediction errors, relative to equivalent census variables, than the separable model and the state-of-the-art multivariate model with unstructured dependence between outcomes and no spatial dependence. In addition, both the GMCAR and the separable models give smaller mean squared prediction error than the state-of-the-art model when conducting small area estimation on county level data from the American Community Survey.

Approximate Bayesian computation (ABC) is an approach to sampling from an approximate posterior distribution in the presence of a computationally intractable likelihood function. A common implementation is based on simulating model, parameter and dataset triples from the prior, and then accepting as samples from the approximate posterior, those model and parameter pairs for which the corresponding dataset, or a summary of that dataset, is ‘close’ to the observed data. Closeness is typically determined though a distance measure and a kernel scale parameter. Appropriate choice of that parameter is important in producing a good quality approximation. This paper proposes diagnostic tools for the choice of the kernel scale parameter based on assessing the coverage property, which asserts that credible intervals have the correct coverage levels in appropriately designed simulation settings. We provide theoretical results on coverage for both model and parameter inference, and adapt these into diagnostics for the ABC context. We re-analyse a study on human demographic history to determine whether the adopted posterior approximation was appropriate. Code implementing the proposed methodology is freely available in the R package abctools.

Investigators often gather longitudinal data to assess changes in responses over time within subjects and to relate these changes to within-subject changes in predictors. Missing data are common in such studies and predictors can be correlated with subject-specific effects. Maximum likelihood methods for generalized linear mixed models provide consistent estimates when the data are ‘missing at random’ (MAR) but can produce inconsistent estimates in settings where the random effects are correlated with one of the predictors. On the other hand, conditional maximum likelihood methods (and closely related maximum likelihood methods that partition covariates into between- and within-cluster components) provide consistent estimation when random effects are correlated with predictors but can produce inconsistent covariate effect estimates when data are MAR. Using theory, simulation studies, and fits to example data this paper shows that decomposition methods using complete covariate information produce consistent estimates. In some practical cases these methods, that ostensibly require complete covariate information, actually only involve the observed covariates. These results offer an easy-to-use approach to simultaneously protect against bias from both cluster-level confounding and MAR missingness in assessments of change.

Adaptive cluster sampling can be a useful design for surveying rare and clustered populations. Here we present a new development in adaptive cluster sampling where we use a two-stage design and extend the complete allocation sampling method. In the proposed new design the primary sample units are selected and, depending on the value of a preset condition, the entire primary unit is surveyed, as in complete allocation sampling. In the next step, if a second condition is met, the surrounding primary sample units are selected. We review the efficiency of the proposed design for sampling the New Zealand Castle Hill buttercups and provide unbiased estimators for the population total and sampling variance.

When testing treatment effects in multi-arm clinical trials, the Bonferroni method or the method of Simes 1986) is used to adjust for the multiple comparisons. When control of the family-wise error rate is required, these methods are combined with the close testing principle of Marcus *et al*. (1976). Under weak assumptions, the resulting *p*-values all give rise to valid tests provided that the basic test used for each treatment is valid. However, standard tests can be far from valid, especially when the endpoint is binary and when sample sizes are unbalanced, as is common in multi-arm clinical trials.

This paper looks at the relationship between size deviations of the component test and size deviations of the multiple comparison test. The conclusion is that multiple comparison tests are as imperfect as the basic tests at nominal size *α*/*m* where *m* is the number of treatments. This, admittedly not unexpected, conclusion implies that these methods should only be used when the component test is very accurate at small nominal sizes. For binary end-points, this suggests use of the parametric bootstrap test. All these conclusions are supported by a detailed numerical study.

Consider a linear regression model with independent normally distributed errors. Suppose that the scalar parameter of interest is a specified linear combination of the components of the regression parameter vector. Also suppose that we have uncertain prior information that a parameter vector, consisting of specified distinct linear combinations of these components, takes a given value. Part of our evaluation of a frequentist confidence interval for the parameter of interest is the scaled expected length, defined to be the expected length of this confidence interval divided by the expected length of the standard confidence interval for this parameter, with the same confidence coefficient. We say that a confidence interval for the parameter of interest utilizes this uncertain prior information if (a) the scaled expected length of this interval is substantially less than one when the prior information is correct, (b) the maximum value of the scaled expected length is not too large and (c) this confidence interval reverts to the standard confidence interval, with the same confidence coefficient, when the data happen to strongly contradict the prior information. We present a new confidence interval for a scalar parameter of interest, with specified confidence coefficient, that utilizes this uncertain prior information. A factorial experiment with one replicate is used to illustrate the application of this new confidence interval.

In this paper, we consider some results on distribution theory of multivariate progressively Type-II censored order statistics. We also establish some characterizations of Freund's bivariate exponential distribution based on the lack of memory property.

By means of a search design one is able to search for and estimate a small set of non-zero elements from the set of higher order factorial interactions in addition to estimating the lower order factorial effects. One may be interested in estimating the general mean and main effects, in addition to searching for and estimating a non-negligible effect in the set of 2- and 3-factor interactions, assuming 4- and higher-order interactions are all zero. Such a search design is called a ‘main effect plus one plan’ and is denoted by MEP.1. Construction of such a plan, for 2* ^{m}* factorial experiments, has been considered and developed by several authors and leads to MEP.1 plans for an odd number

In this paper the issue of finding uncertainty intervals for queries in a Bayesian Network is reconsidered. The investigation focuses on Bayesian Nets with discrete nodes and finite populations. An earlier asymptotic approach is compared with a simulation-based approach, together with further alternatives, one based on a single sample of the Bayesian Net of a particular finite population size, and another which uses expected population sizes together with exact probabilities. We conclude that a query of a Bayesian Net should be expressed as a probability embedded in an uncertainty interval. Based on an investigation of two Bayesian Net structures, the preferred method is the simulation method. However, both the single sample method and the expected sample size methods may be useful and are simpler to compute. Any method at all is more useful than none, when assessing a Bayesian Net under development, or when drawing conclusions from an ‘expert’ system.