Estimation of extreme values index in the presence of censored data Case study : Unemployment durations in Algeria

Abstract

Extreme events, generally catastrophic (earthquakes, floods, nuclear accidents, monetary or financial crises, stock market crashes, emergence of a new endemic phenomenon, etc.) dominate the daily news by their unpredictability. Given the importance of social and economic consequences, no serious debate on hazard can be conducted without a reflection on rare and extreme events. The use of the laws of extreme values, which corresponds to the study of the distribution tail of a phenomenon, is based on properties of order statistics and on extrapolation methods. More precisely, on the convergences in law of the maxima of appropriately normalized random variables. At the statistical level, the difficulties arise especially when one wants to model events so rare that they have never been observed, since the most relevant information concerning these unobserved extreme values is contained in the most extreme values observed. However, in practice, especially in applications and statistical analyzes that requires a long series or a large database, have always been faced with the problems of censored values, ie incomplete data that is equivalent to a lack of observation or a loss of information about the event studied . This poses a real problem for practitioners even in the normal case, let alone the case of extreme values which are by definition events with a low probability of occurrence, as these are much larger values or smaller than those usually observed. Recently, many researchers are interested in and orientate towards this line of research, a relatively new line of research that constitutes a kind of marriage between two branches of statistics: the theory of extreme values and survival analysis. To this end, our goal throughout this work was to try to make a small contribution to improve the performance and quality of the extreme index estimator, when the data is submitted to random censorship. For this purpose, an extreme tail index estimator taking into account the presence of randomly censored data on the right has been proposed by maximizing the likelihood of EVT under random censoring by the Newton Raphson algorithm, where his behavior was examined and evaluated on simulated data. The results obtained are then applied to real data of the duration of unemployment in Algeria.