Francesco Pauli

Penalized likelihood inference in extreme value analyses

Models for extreme values are usually based on detailed asymptotic argument, for which strong ergodic assumptions such as stationarity, or prescribed perturbations from stationarity, are required. In most applications of extreme value modelling such assumptions are not satisfied, but the type of departure from stationarity is either unknown or complex, making asymptotic calculations unfeasible. This has led to various approaches in which standard extreme value models are used as building blocks for conditional or local behaviour of processes, with more general statistical techniques being used at the modelling stage to handle the non-stationarity. This paper presents another approach in this direction based on penalized likelihood. There are some advantages to this particular approach: the method has a simple interpretation; computations for estimation are relatively straightforward using standard algorithms; and a simple reinterpretation of the model enables broader inferences, such as confidence intervals, to be obtained using MCMC methodology. Methodological details together with applications to both athletics and environmental data are given.

Climate Change and Extreme Events: An Assessment of Economic Implications

We use a general equilibrium model of the world economy, and a regional economic growth model, to assess the economic implications of vulnerability from extreme meteorological events, induced by the climate change. In particular, we first consider the impact of climate change on ENSO and NAO oceanic oscillations and, subsequently, the implied variation on regional expected damages. We found that expected damages from extreme events are increasing in the United States, Europe and Russia, and Russia, and decreasing in energy exporting countries. Two economic implications are taken into account: (1) short-term impacts, due to changes in the demand structure, generated by higher/lower precautionary saving, and (2) variations in regional economic growth paths. We found that indirect short-term effects (variations in savings due to higher or lower likelihood of natural disasters) can have an impact on regional economics, whose order of magnitude is comparable to the one of direct damages. On the other hand, we highlight that higher vulnerability from extreme events translates into higher volatility in the economic growth path, and vice versa.

Smoothing sample extremes: The mixed model approach

Nonparametric regression for sample extremes can be performed using a variety of techniques. The penalized spline approach for the Poisson point process model is considered. The generalized linear mixed model representation for the spline model, with its Bayesian approach to inference, turns out to be a very flexible framework. Monte Carlo Markov chain algorithms are employed for exploration of the posterior distribution. The overall performance of the method is tested on simulated data. Two real data applications are also discussed for modeling trend of intensity of earthquakes in Italy and for assessing seasonality and short term trend of summer extreme temperatures in Milan, Italy.

Extremal limit laws for a class of bivariate Poisson vectors

It is well known that conventional extreme value limit laws break down for the Poisson distribution: no normalization can be found to avoid degeneracy of the limit law of sample maxima. Anderson et al. (Ann. Appl. Probab. 7 (1997) 953) tackled this problem with a triangular array argument, letting both the sample size and Poisson mean grow at appropriate rates. This leads to a Gumbel limit law for sample maxima. In applications, this means that it may be appropriate to model extremes of Poisson processes using standard extreme value models and techniques. This paper extends the limit results to a class of bivariate Poisson distributions. Suitably normalized, and with a degree of dependence that is also permitted to grow at a suitable rate, we find that the limit distribution corresponds to the class of bivariate extreme value models that would have arisen, had the population been bivariate normal, cf. Husler and Reiss (Statist. Probab. Lett. 7 (1989) 283). This adds weight to the argument that, for practical applications involving Poisson variables, even in the presence of dependence, standard extreme value models can be applied, despite the degeneracy that arises by applying the usual asymptotic argument.

A note on empirical likelihoods derived from pairwise score functions

Pairwise likelihood functions are convenient surrogates for the ordinary likelihood, useful when the latter is too difficult or even impractical to compute. One drawback of pairwise likelihood inference is that, for a multidimensional parameter of interest, the pairwise likelihood analogue of the likelihood ratio statistic does not have the standard chi-square asymptotic distribution. Invoking the theory of unbiased estimating functions, this paper proposes and discusses a computationally and theoretically attractive approach based on the derivation of empirical likelihood functions from the pairwise scores. This approach produces alternatives to the pairwise likelihood ratio statistic, which allow reference to the usual asymptotic chi-square distribution and which are useful when the elements of the Godambe information are troublesome to evaluate or in the presence of large data sets with relative small sample sizes. Two Monte Carlo studies are performed in order to assess the finite-sample performance of the proposed empirical pairwise likelihoods.

Modelling short-term effects of ozone on morbidity: an application to the city of Milano, Italy, 1995–2003

In this paper, we explore a range of concerns that arise in measuring short-term effects of ozone on health. In particular, we tackle the problem of measuring exposure using alternative daily measures of ozone derived from hourly concentrations. We adopt the exposure paradigm of Chiogna and Bellini (Environmetrics 13:55–69, 2002) extending it to ozone concentrations, and we compare its performances with respect to traditional exposure measures by exploiting model selection. To investigate the stability of model selection, we then apply the idea of bootstrapping the modelling process.

Summer temperature effects on deaths and hospital admissions among the elderly population in two Italian cities

In developed countries the effects of climate on health status are mainly due to temperature. Our analysis is aimed to deepen statistically the relationship between summer climate conditions and daily frequency of health episodes: deaths or hospital admissions. We expect to find a U-shaped relationship between temperature and frequencies of events occurring in summer regarding the elderly population resident in Milano and Brescia. We use as covariates hourly records of temperature recorded at observation sites located in Milano and Brescia. The analysis is performed using Generalized Additive Models (GAM), where the response variable is the daily number of events, which varies as a possibly non-linear function of meteorological variables measured on the same or previous day. We consider separate models for Milano and Brescia and then we compare temperature effects among the two towns and among different age classes. Moreover we consider separate models for all diagnosed events, for those due to respiratory disease and those due to circulatory pathologies. Model selection is a central problem, the basic methods used are the UBRE and GCV criteria but, instead of conditioning all final conclusions on the best model according to the chosen criterion, we investigated the effect of model selection by implementing a bootstrap procedure.