Lászlό Márkus

The talk is intended to give a brief survey of our modelling efforts of Rivers Tisza and Danube in Hungary. Daily river discharge data registered in the last century is the basis of our analysis. Our study is motivated mainly by flood risk estimation, hence, model construction has to focus on extremes, and be tested against the fit of quantiles (return levels), maxima, extremal index (clustering of high flows), time spent over threshold (flood duration) and aggregate excesses (flood volume). To obtain reliable information on the returns of floods the dynamics of the flows and the interdependence of tributaries with the main river have to be represented properly.

The deseasonalised river flow series has skewed, leptokurtic and light-tailed marginal distribution, correlated squares and clustering level exceedances. As the fit of conventional GARCH or ARMA-GARCH or LARCH models is unsatisfactory in terms of extremal behaviour, we suggest conditioning the variance on a direct feedback from the modelled flows. The linear form of the conditional variance creates moment equivalence between the modelled process and the generating noise. Further, when a simplified model is driven by a Weibull-type noise, the generated process has approximately Weibull like tail, too, albeit with different exponent: 1/2 that of the noise. The observed data supports this conjecture for the more complex model as well.

As a proper representation of the lagged interdependence structure is also an important aspect of the goodness of fit, we compare the model time series by a novel approach. The idea is to extend the use of copulas to the lagged (one-dimensional) series, to the analogy of the autocorrelation function. The advantage of the use of such autocopulas lays in the fact that they represent nonlinear dependencies as well; hence they can reveal the specifics of the lagged interdependence in a much finer way. True though, it seems rather cumbersome to calculate the exact form of the autocopula even for the simplest nonlinear time series models, therefore, we confine ourselves to an empirical and simulation based approach. Tests based on Kendall's transform will be presented in order to check whether observed and model autocopulas can be distinguished significantly.

Finally, I will mention some aspects of simultaneous modelling of the main river and its tributaries.

**Keywords:** Copulas; Extreme values; Non-linear time series; River discharge series

**Biography:** Laszlo Markus is professor of statistics at the Eotvos Lorand University, Budapest, Hungary. He is a recently elected member of ISI, and currently chairs the European Regional Committee of the Bernoulli Society.