We describe a class of random field models for geostatistical count data based on Gaussian copulas. Unlike hierarchical Poisson models often used to describe this type of data, Gaussian copula models allow a more direct modelling of the marginal distributions and association structure of the count data. We study in detail the correlation structure of these random fields when the family of marginal distributions is either negative binomial or zero-inflated Poisson; these represent two types of overdispersion often encountered in geostatistical count data. We also contrast the correlation structure of one of these Gaussian copula models with that of a hierarchical Poisson model having the same family of marginal distributions, and show that the former is more flexible than the latter in terms of range of feasible correlation, sensitivity to the mean function and modelling of isotropy. An exploratory analysis of a dataset of Japanese beetle larvae counts illustrate some of the findings. All of these investigations show that Gaussian copula models are useful alternatives to hierarchical Poisson models, specially for geostatistical count data that display substantial correlation and small overdispersion.

Relative risks are often considered preferable to odds ratios for quantifying the association between a predictor and a binary outcome. Relative risk regression is an alternative to logistic regression where the parameters are relative risks rather than odds ratios. It uses a log link binomial generalised linear model, or log-binomial model, which requires parameter constraints to prevent probabilities from exceeding 1. This leads to numerical problems with standard approaches for finding the maximum likelihood estimate (MLE), such as Fisher scoring, and has motivated various non-MLE approaches. In this paper we discuss the roles of the MLE and its main competitors for relative risk regression. It is argued that reliable alternatives to Fisher scoring mean that numerical issues are no longer a motivation for non-MLE methods. Nonetheless, non-MLE methods may be worthwhile for other reasons and we evaluate this possibility for alternatives within a class of quasi-likelihood methods. The MLE obtained using a reliable computational method is recommended, but this approach requires bootstrapping when estimates are on the parameter space boundary. If convenience is paramount, then quasi-likelihood estimation can be a good alternative, although parameter constraints may be violated. Sensitivity to model misspecification and outliers is also discussed along with recommendations and priorities for future research.

The recent blistering heat waves of 2009 in the state of Victoria in Australia were so unprecedented in terms of duration and intensity that society was largely unprepared. These heat waves caused serious health, social and economic problems. In this paper, the daily maximum temperatures at ten selected stations are studied. Auto-regressive integrated moving-average models are used to prewhiten the time series. Uncorrelated, non-normal and heavy-tailed residuals are analyzed by means of a new skew *t*-mixture distribution. The number of mixture components is effectively determined by an innovative penalisation procedure. It is shown that the resulting skew *t*-mixture models provide an acceptable fit in all cases. Possible future temperature patterns are obtained through simulation. It is forecast that the average duration of high temperature episodes will increase by two to three days per year and a new eight-year high temperature level is very likely in the coming few years. The relationship between heavy tail behaviour of the fitted distribution and heat waves is noteworthy.

In this paper we present a semiparametric test of goodness of fit which is based on the method of L-moments for the estimation of the nuisance parameters. This test is particularly useful for any distribution that has a convenient expression for its quantile function. The test proceeds by investigating equality of the first few L-moments of the true and the hypothesised distributions. We provide details and undertake simulation studies for the logistic and the generalised Pareto distributions. Although for some distributions the method of L-moments estimator is less efficient than the maximum likelihood estimator, the former method has the advantage that it may be used in semiparametric settings and that it requires weaker existence conditions. The new test is often more powerful than competitor tests for goodness of fit of the logistic and generalised Pareto distributions.

Group testing is the process of combining individual samples and testing them as a group for the presence of an attribute. The use of such testing to estimate proportions is an important statistical tool in many applications. When samples are collected and tested in groups of different size, complications arise in the construction of exact confidence intervals. In this case, the numbers of positive groups has a multivariate distribution, and the difficulty stems from a lack of a natural ordering of the sample points. Exact two-sided intervals such as the equal-tail method based on maximum likelihood estimation, and those based on joint probability or likelihood ratio statistics, have been previously considered. In this paper several new estimators are developed and assessed. We show that the combined tails (or Blaker) method based on a suitable ordering statistic, is the best choice in this setting. The methods are illustrated using a study involving the infection prevalence of *Myxobolus cerebralis* among free-ranging fish.

The paper considers a model for crossover designs with carryover effects and a random interaction between treatments and subjects. Under this model, two observations of the same treatment on the same subject are positively correlated and therefore provide less information than two observations of the same treatment on different subjects. The introduction of the interaction makes the determination of optimal designs much harder than is the case for the traditional model. Generalising the results of Bludowsky's thesis, the present paper uses Kushner's method to determine optimal approximate designs. We restrict attention to the case where the number of periods is less than or equal to the number of treatments. We determine the optimal designs in the important special cases that the number of periods is 3, 4 or 5. It turns out that the optimal designs depend on the variance of the random interactions and in most cases are not binary. However, we can show that neighbour balanced binary designs are highly efficient, regardless of the number of periods and of the size of the variance of the interaction effects.

Mood's test, which is a relatively old test (and the oldest non-parametric test among those tests in its class) for determining heterogeneity of variance, is still being widely used in different areas such as biometry, biostatistics and medicine. Although it is a popular test, it is not suitable for use on a two-way factorial design. In this paper, Mood's test is generalised to the 2 × 2 factorial design setting and its performance is compared with that of Klotz's test. The power and robustness of these tests are examined in detail by means of a simulation study with 10,000 replications. Based on the simulation results, the generalised Mood's and Klotz's tests can especially be recommended in settings in which the parent distribution is symmetric. As an example application we analyse data from a multi-factor agricultural system that involves chilli peppers, nematodes and yellow nutsedge. This example dataset suggests that the performance of the generalised Mood test is in agreement with that of the generalised Klotz's test.

Informative identification of the within-subject correlation is essential in longitudinal studies in order to forecast the trajectory of each subject and improve the validity of inferences. In this paper, we fit this correlation structure by employing a time adaptive autoregressive error process. Such a process can automatically accommodate irregular and possibly subject-specific observations. Based on the fitted correlation structure, we propose an efficient two-stage estimator of the unknown coefficient functions by using a local polynomial approximation. This procedure does not involve within-subject covariance matrices and hence circumvents the instability of calculating their inverses. The asymptotic normality of resulting estimators is established. Numerical experiments were conducted to check the finite sample performance of our method and an example of an application involving a set of medical data is also illustrated.