Estimation of spatial max-stable models using threshold exceedances

Parametric inference for spatial max-stable processes is difficult since the related likelihoods are unavailable. A composite likelihood approach based on the bivariate distribution of block maxima has been recently proposed. However modeling block maxima is a wasteful approach provided that other information is available. Moreover an approach based on block maxima, typically annual, is unable to take into account the fact that maxima occur or not simultaneously. If time series of, say, daily data are available, then estimation procedures based on exceedances of a high threshold could mitigate such problems. We focus on two approaches for composing likelihoods based on pairs of exceedances. The first one comes from the tail approximation for bivariate distribution proposed by Ledford and Tawn (Biometrika 83:169–187, 1996) when both pairs of observations exceed the fixed threshold. The second one uses the bivariate extension (Rootzén and Tajvidi in Bernoulli 12:917–930, 2006) of the generalized Pareto distribution which allows to model exceedances when at least one of the components is over the threshold. The two approaches are compared through a simulation study where both processes in a domain of attraction of a max-stable process and max-stable processes are successively considered as time replications, according to different degrees of spatial dependency. Results put forward how the nature of the time replications influences the bias of estimations and highlight the choice of each approach regarding to the strength of the spatial dependencies and the threshold choice.     

The Onset of India's Fertility Transition

The all-India trend rate of fertility change, since 1961, was obtained from kriged maps (reconstituted surfaces) of a fertility index drawn from the data of the four decennial censuses conducted between 1961 and 1991, at the district level. This rate is calculated as the relative variation of the fertility index between consecutive censuses. It actually represents the change related to any measurement of fertility such as the Total Fertility Rate (TFR). Based on the surfaces of change, the onset of the fertility transition was estimated by using an auxiliary variable which takes into account both the random fluctuations in the pre-transition change at the district level and the absorbing state which constitutes the transition. The space-time analysis (Mantel test) of transition through this auxiliary variable shows aggregates of districts in transition at short geographical distances, but no geographical diffusion – neither on a sub-continental scale nor in the three selected regions having high and low fertility. This represents a process of vertical, but non-social, diffusion set in motion by the state machinery called a ``top-down' process (Srinivasan, 1995). This top-down process does not resemble the transition diffusion at its onset in Europe which had a strong geographical component.     

Estimating and testing zones of abrupt change for spatial data

We propose a method for detecting the zones where a variable irregularly sampled in the plane changes abruptly. Our general model is that under the null hypothesis the variable is the realisation of a stationary Gaussian process with constant expectation. The alternative is that the mean function presents abrupt changes. We define potential Zones of Abrupt Change (ZACs) by the points where the gradient, estimated under the null hypothesis, exceeds a determined threshold. We then design a global test to assess the global significance of the potential ZACs, an issue missing in all existing methods. The theory that links the threshold and the global level is based on asymptotic distributions of excursion sets of non-stationary χ ² fields for which we provide new results. The method is evaluated by a simulation study and applied to a soil data set in the context of precision agriculture.