# ENBIS: European Network for Business and Industrial Statistics

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## ENBIS-17 in Naples

*9 – 14 September 2017; Naples (Italy)*Abstract submission: 21 November 2016 – 10 May 2017

### Distributions for the Maximum of a Random Number of Observations

*11 September 2017, 16:40 – 17:00*

#### Abstract

- Submitted by
- Marta Perez-Casany
- Authors
- Marta Perez-Casany (Universitat Politècnica de Catalunya)
- Abstract
- Extreme values theory is the branch of statistics that deal with the problem of modelling and predicting events that are far away from the its usual range. It is required in many research areas as industry (for predicting important system malfunctions), in insurance ( for predicting huge number of claims due to catastrophic events), in environmental studies ( for predicting excess of contamination in big cities), etc. This theory studies the distribution of maximums, events that are much larger than the usual observations, and its main theorem, known as Fisher-Tippet-Gnedenko theorem, establishes that the maximum follows asymptotically a Generalized Extreme Value (GEV) distribution, but the convergence is, in general, slow. Another important theorem is the one due to Pickards, Balkema and de Haan that states that the observations over a threshold have as asymptotic distribution a Generalized Pareto distribution (GP).

In many practical cases, one is interested in the maximum (minimum) value of a given measure during a given period of time or space, without knowing in advance how many observations there will be. In that case one needs to consider maximums (minimums) of a random number of independent copies of a given random variable, which is denoted as random stopped extreme distributions (RSED). The RSED depends on the distribution of the phenomena and on the distribution of the number of events. The main advantage of using RSED is that it does not require to have a large number of observations. The classical theory of Extreme values may be seen as a particular case. Emphasis will be given to the particular case where the number of observations follows a Positive Poisson distribution.