Extreme value statistics Density of near-extreme events Sanjib Sabhapandit Laboratoire de Physique Th´ eorique et Mod` eles Statistiques CNRS UMR 8626 — Universit´ e Paris-Sud 91405 Orsay cedex, France Collaborator Ref. Phys. Rev. Lett. 98 , 140201 (2007). Satya N. Majumdar
Extreme value statistics : The statistics of the maximum or the minimum value of a set of random observations { X 1 , X 2 , . . . , X N } . S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 1 / 26
Outline 1 a brief historical introduction to the extreme value statistics extreme value statistics of i.i.d. random variables 2 ◮ the three limiting distributions ◮ their appearence in other problems extreme value statistics in weakly correlated variables 3 near-extreme events 4 ◮ density of states with respect to the extreme value ◮ limiting behavior of the mean density of states ◮ illustration with explicit examples ◮ comparison with Yamal summer temperature data summary 5 S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 2 / 26
A brief history of extreme value statistics Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t . Ans. t N / ( N + 1 ) ( see [Gumbel (1958)] ) S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 3 / 26
A brief history of extreme value statistics Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t . Ans. t N / ( N + 1 ) ( see [Gumbel (1958)] ) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range ( = max − min) of random samples from the Gaussian distribution. [von Bortkiewicz (1922)] —introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 3 / 26
A brief history of extreme value statistics Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t . Ans. t N / ( N + 1 ) ( see [Gumbel (1958)] ) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range ( = max − min) of random samples from the Gaussian distribution. [von Bortkiewicz (1922)] —introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] Richard von Mises (1923): The expected value. [von Mises (1923)] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 3 / 26
A brief history of extreme value statistics Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t . Ans. t N / ( N + 1 ) ( see [Gumbel (1958)] ) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range ( = max − min) of random samples from the Gaussian distribution. [von Bortkiewicz (1922)] —introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] Richard von Mises (1923): The expected value. [von Mises (1923)] Edward Lewis Dodd (1923): The median. [Dodd (1923)] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 3 / 26
A brief history of extreme value statistics Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t . Ans. t N / ( N + 1 ) ( see [Gumbel (1958)] ) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range ( = max − min) of random samples from the Gaussian distribution. [von Bortkiewicz (1922)] —introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] Richard von Mises (1923): The expected value. [von Mises (1923)] Edward Lewis Dodd (1923): The median. [Dodd (1923)] Maurice Ren´ e Fr´ echet (1927): Obtained two of the three kinds of extreme-value distribution. echet (1927)] (not printed until 1928) [Fr´ S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 3 / 26
A brief history of extreme value statistics Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t . Ans. t N / ( N + 1 ) ( see [Gumbel (1958)] ) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range ( = max − min) of random samples from the Gaussian distribution. [von Bortkiewicz (1922)] —introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] Richard von Mises (1923): The expected value. [von Mises (1923)] Edward Lewis Dodd (1923): The median. [Dodd (1923)] Maurice Ren´ e Fr´ echet (1927): Obtained two of the three kinds of extreme-value distribution. echet (1927)] (not printed until 1928) [Fr´ Sir Ronald Aylmer Fisher & Leonard Henry Caleb Tippett (1928): Obtained three types of distributions and showed that extreme limit distributions can only be one of the three types. [Fisher & Tippett (1928)] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 3 / 26
A brief history of extreme value statistics Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t . Ans. t N / ( N + 1 ) ( see [Gumbel (1958)] ) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range ( = max − min) of random samples from the Gaussian distribution. [von Bortkiewicz (1922)] —introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] Richard von Mises (1923): The expected value. [von Mises (1923)] Edward Lewis Dodd (1923): The median. [Dodd (1923)] Maurice Ren´ e Fr´ echet (1927): Obtained two of the three kinds of extreme-value distribution. echet (1927)] (not printed until 1928) [Fr´ Sir Ronald Aylmer Fisher & Leonard Henry Caleb Tippett (1928): Obtained three types of distributions and showed that extreme limit distributions can only be one of the three types. [Fisher & Tippett (1928)] Boris Vladimirovich Gnedenko (1943): Provided rigorous foundation and necessary and sufficient conditions for the weak convergence to the extreme limit distributions. [Gnedenko (1943)] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 3 / 26
A brief history ... (II): applications Emil Julius Gumbel [Gumbel (1958)] ◮ Les intervalles extrˆ emes entre les ´ emissions radio-actives. (Extreme intervals between radioactive emissions) J. Phys. Radium. 8 , 321-329, (1937). ◮ Les intervalles extrˆ emes entre les ´ emissions radioactives. II J. Phys. Radium 8 , 446-452 (1937) ◮ The return period of flood flows. Ann. Math. Statistics 12 , 163–190 (1941). ◮ Probability-interpretation of the observed return-periods of floods. Trans. Amer. Geophys. Union 1941 , 836–850 (1941). ◮ On the frequency distribution of extreme values in meteorological data. Bull. Amer. Meteorol. Soc. 23 , 95–105 (1942). ◮ Statistical forecast of droughts. Bull. Int. Assoc. Sci. Hydrol. 8 , 5-23 (1963). S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 4 / 26
A brief history ... (II): applications Emil Julius Gumbel [Gumbel (1958)] ◮ Les intervalles extrˆ emes entre les ´ emissions radio-actives. (Extreme intervals between radioactive emissions) J. Phys. Radium. 8 , 321-329, (1937). ◮ Les intervalles extrˆ emes entre les ´ emissions radioactives. II J. Phys. Radium 8 , 446-452 (1937) ◮ The return period of flood flows. Ann. Math. Statistics 12 , 163–190 (1941). ◮ Probability-interpretation of the observed return-periods of floods. Trans. Amer. Geophys. Union 1941 , 836–850 (1941). ◮ On the frequency distribution of extreme values in meteorological data. Bull. Amer. Meteorol. Soc. 23 , 95–105 (1942). ◮ Statistical forecast of droughts. Bull. Int. Assoc. Sci. Hydrol. 8 , 5-23 (1963). Wallodi Weibull : ◮ A statistical theory of the strength of material (transl.). Ingvetensk. Akad. Handl 151 , 1-45 (1939). ◮ A statistical distribution function of wide applicability. J. Appl. Mech. 18 , 293-277 (1951). 1. Yield strength of a Bofors steel. 2. Size distribution of fly ash. 3. Fiber strength of Indian cotton. 4. Length of cytoidea (Worm length for ancient sedimentary deposits). 5. Fatigue life of a St-37 steel. 6. Statures for adult males, born in the British Isles. 7. Breadth of beans of Phaseolux Vulgaris. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 4 / 26
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