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In this paper, we consider the problem of specifying priors for the variance components in the Bayesian analysis of the Besag-York-Mollié model, a model that is popular among epidemiologists for disease mapping. The model encompasses two sets of random effects: one spatially structured to model spatial autocorrelation and the other spatially unstructured to describe residual heterogeneity. In this model, prior specification for variance components is an important problem because these priors maintain their influence on the posterior distributions of relative risks when mapping rare diseases. We propose using generalised inverse Gaussian priors, a broad class of distributions that includes many distributions commonly used as priors in this context, such as inverse gammas. We discuss the prior parameter choice with the aim of balancing the prior weight of the two sets of random effects on total variation and controlling the amount of shrinkage. The suggested prior specification strategy is compared to popular alternatives using a simulation exercise and applications to real data sets., Statistica, Vol 76, No 1 (2016)

La stima dell’indice di concentrazione di Gini per il reddito, i consumi o altre variabili economiche a livello di piccole aree `e uno strumento utile per analizzare il livello di disuguaglianza all’interno di territori circoscritti. La letteratura sulla stima per piccole aree che riguarda questo indicatore `e tuttavia limitata e basata su un’impostazione i cui presupposti non sono soddifatti in molti contesti concreti di stima. In questo lavoro proponiamo un metodo di stima per piccole aree basato su una modellazione a livello di area. Poich´e questo metodo presuppone che lo stimatore diretto sia approssimativamente corretto, proponiamo una correzione dello stimatore da disegno dell’indice di Gini in domini di studio che ne riduce la distorsione nei piccoli domini. Il modello di piccole aree che illustriamo utilizzando dati dell’indagine EU-SILC `e una regressione mista basata su una verosimiglianza. Beta. L’inferenza sul modello `e condotta secondo un’impostazione Bayesiana The Gini coefficient is a popular concentration measure often used in the analysis of economic inequality. Estimates of this index for small regions may be useful to properly represent inequalities within local communities. The literature on small area estimation for the Gini coefficient is limited. In this paper, we propose a method based on area-level models, thereby not assuming the availability of Census data at the micro-level. We propose a modified design-based estimator for the coefficient with reduced small sample bias and we consider a hierarchical Beta mixed regression model to combine survey data and auxiliary information. The methodology is illustrated by means of an example based on Italian data from the EU-SILC survey.

Disease mapping studies have been widely performed at univariate level, that is considering only one disease in the estimated models. Nonetheless, simultaneous modeling of different diseases can be a valuable tool both from the epidemiological and from the statistical point of view. In this paper we propose a model for bivariate disease mapping that generalises the univariate CAR distribution. The proposed model is proven to be an effective alternative to existing bivariate models, mainly because it overcome some restrictive hypotheses underlying models previously proposed in this context. Model performances are checked via a simulation study and via application to some real case studies., Statistica; Vol 68, No 3/4 (2008); 327-347

PM10 monitoring networks are equipped with heterogeneous samplers. Some of these samplers are known to underestimate true levels of concentrations (non-reference samplers). In this paper we propose a hierarchical spatio-temporal Bayesian model for the calibration of measurements recorded using non-reference samplers, by borrowing strength from non co-located reference sampler measurements., Statistica; Vol 66, No 2 (2006); 127-138

The lognormal distribution is a popular model in many fields of statistics. The mean, the mode, the median and other summaries of the lognormal may be written as special cases of θa,b = exp(aξ + bσ2) for different choices of a, b. Bayesian inference on θa,b under quadratic loss is problematic since, for many popular choices of the prior for σ2, it has no finite moments. In this paper we propose a Generalized Inverse Gaussian prior for σ2 that leads to a log-Generalized Hyperbolic posterior for θa,b, a distribution for which it is easy to calculate quantiles and moments, provided that they exist. We derive the constraints on the prior parameters that yields finite posterior moments of order r and closed form expressions for the posterior mean and variance of θa,b. We investigate the choice of prior parameters leading to a posterior mean with optimal frequentist MSE using a second-order 'small argument' approximation to the modified Bessel functions of the third kind. We also explore prior specification when the hypotheses on which the 'small argument' approximation is based do not hold. The relative quadratic loss function has often been used in the literature to summarize the posterior for θa,b. We show that, under our approach, also the Bayes estimator obtained minimizing the relative quadratic loss may be written in closed form. With reference to the estimation of the lognromal mean we show that popular estimators may be obtained as special cases of our posterior mean; we also show, by simulation, that our posterior mean compares favourably to known estimators of the lognromal mean in terms of frequentist MSE. In particular it is more efficient than the Bayes estimator obtained minimizing the relative quadratic loss.

In this work we study sensitivity to hyperprior specification of the dispersion parameter in a convolution model widely used in spatial disease mapping, that accounts for both structured (spatial) and unstructured heterogeneity. In the fully Bayesian approach to disease mapping, hyperprior choice sensibly affects inferences. In this work we critically review the most common hyperprior specifications. Moreover we propose a new hyperprior distribution, the Generalised Inverse Gaussian, starting from an idea explored in the estimation of the mean for iid log-Normal observations. In this context it is well known that hyperprior parameters have to be set accurately in order to avoid infinite moments of the posterior distribution. The performances of various hyperprior choices are compared via simulation studies. In particular we investigate the sensitivity of the relative risk estimates.

The log-normal distribution is a popular model in biostatistics as in many other fields of statistics. Bayesian inference on the mean and median of the distribution is problematic because, for many popular choices of the prior for variance (on the log-scale) parameter, the posterior distribution has no finite moments, leading to Bayes estimators with infinite expected loss for the most common choices of the loss function. In this paper we propose a generalized inverse Gaussian prior for the variance parameter, that leads to a log-generalized hyperbolic posterior, a distribution for which it is easy to calculate quantiles and moments, provided that they exist. We derive the constraints on the prior parameters that yields finite posterior moments of order r. For the quadratic and relative quadratic loss functions, we investigate the choice of prior parameters leading to Bayes estimators with optimal frequentist mean square error. For the estimation of the lognormal mean we show, using simulation, that the Bayes estimator under quadratic loss compares favorably in terms of frequentist mean square error to known estimators. The theory does not apply only to the mean or median estimation but to all parameters that may be written as the exponential of a linear combination of the distribution’s two parameters that include the mode and all non central moments.

The Italian National Institute of Statistics (Istat) intends to collect some required information for the 2011 Population Census using a sample survey. Specifically, de-tailed information from large municipalities (with a population over 20,000 inhabit-ants) will be collected from a sample of the population using a long-form question-naire. For the rest of the population, partial data, consisting mainly of demographic variables, will be collected using a short-form questionnaire. Thus, data will be ob-tained from a combination of long and short-form questionnaires. In municipalities with a population under 20,000 inhabitants, a traditional census (based on the long-form) will be carried out. Moreover, in each larger municipality, only census areas with a population over 10,000 inhabitants will be sampled. In terms of the sampling design, simple random sampling (SRS) of households with a sampling fraction of 0.33 from population registers will be adopted (Carbonetti and Fortini, 2008). The innovations produce methodological challenges. For example, estimating a contingency table involving variables from both short and long-form ques-tionnaires is difficult because estimates of the joint distribution must be coherent with the marginal distribution collected by a census. Moreover, several constraints at different levels of geographical aggregation are imposed by the Istat dissemination plan. In a design-based approach, calibration estimation (Deville and Särndal, 1992) is a natural choice for contingency table estimation that comply to such constraints. In this paper, we propose a small area model-based approach that takes into account the coherence constraints and allows borrowing strength between areas to reduce the variance of the estimates.

The topic of the summer school is "Deterministic and Stochastic modelling in Biology and Medicine". Seven specialized lactures were given by national and international experts.

The number of people recruited by firms in Local Labour Market Areas provides an important indicator of the reorganisation of the local productive processes. In Italy, this parameter can be estimated using the information collected in the Excelsior survey, although it does not provide reliable estimates for the domains of interest. In this paper we propose a multivariate small area estimation approach for count data based on the Multivariate Poisson-Log Normal distribution. This approach will be used to estimate the number of firm recruits both replacing departing employees and filling new positions. In the small area estimation framework, it is customary to assume that sampling variances and covariances are known. However, both they and the direct point estimates suffer from instability. Due to the rare nature of the phenomenon we are analysing, counts in some domains are equal to zero, and this produces estimates of sampling error covariances equal to zero. To account for the extra variability due to the estimated sampling covariance matrix, and to deal with the problem of unreasonable estimated variances and covariances in some domains, we propose an “integrated” approach where we jointly model the parameters of interest and the sampling error covariance matrices. We suggest a solution based again on the Poisson-Log Normal distribution to smooth variances and covariances. The results we obtain are encouraging: the proposed small area estimation model shows a better fit when compared to the Multivariate Normal-Normal (MNN) small area model, and it allows for a non-negligible increase in efficiency.

In this paper we study a very basic Small Area model, the Battese-Harter-Fuller, applied to a log-transformed variable and adopt a reference Hierarchical Bayes approach to obtain estimates of finite population Small Area means. With reference Hierarchical Bayesian analysis we mean an analysis in which reference, non-informative priors for unknown parameters are assumed and estimation is implemented by means of Markov Chain Monte Carlo softwares such as OpenBugs (Thomas et al., 2006). One advantage of Hierarchical Bayes approach is that it can easily take into account all sources of variability. This should lead to measure of uncertainty of predicted Small Area means that consider also the variability involved by the estimation of variance components.