What is the chance of an exceptionally serious flood, earthquake, stock market crash or other extreme event? That difficult question to answer is the field of expertise of John Einmahl, professor of statistics in Tilburg and specialist in extreme value theory. On September 8 he held his farewell speech: A beautiful theorem needs no applicationabout the statistics of the exceptional, from athletics records to earthquakes.
How can you say anything with certainty about things that are by definition very rare?
“It is difficult, but it is possible: it is actually a form of extrapolation, based on events that are exceptional, but not extremely exceptional. For example, consider water levels, which you are interested in if you want to build a dike: how high should that dike be to survive a water level that occurs once every ten thousand years?
“Then you start with the highest water levels in a hundred years. These are exceptional, but you have quite a bit of data. And you try to extend that to the more extreme water levels, i.e. the once-in-ten-thousand-year disasters.
“The great thing about statistics is that you can apply the same methods to all kinds of situations: to other disasters, such as earthquakes or floods, but also to financial fluctuations, or to sports records, for example.
“In statistics you always want to reduce variability. In plain English: you want as much certainty as possible and as little uncertainty as possible.
“You can do this, for example, by collecting as much data as possible. When I toss a coin once, I don’t know whether I’m getting heads or tails. But if I do it a million times, I’m pretty sure I’ll hit heads on about half of the throws. That is the law of large numbers.
“But it is not always possible to collect a lot of data, especially for rare events. So the other tactic is to use the data you have as smartly as possible. That is my profession.”
There are physical limits to muscle strength, stride length, etc.: no one will ever run 75 kilometers per hour
‘The devil is in the tail’, I read in a paper about this field.
“Extreme values concern the ‘tail’ of a probability distribution. Look, if this is a probability distribution [hij loopt naar een whiteboard en tekent een grafiek van een berg die naar rechts glooiend afloopt]. This is what happens very often [wijst op de top]these are the events that occur a little less frequently [wijst op de helling]but we are concerned with these values [wijst op de voet van de berg]. These are the extreme, rare events: high water levels, major earthquakes, catastrophic stock market losses. We call that the ‘tail’.
“There are thick and thin tails. An example of a thin tail is the 100 meter sprint, which we have taken into account. In 2009, Usain Bolt ran the world record in the 100 meters, 9.58 seconds. Someone will probably break that record one day, but there are physical limits to muscle strength, stride length, and so on: no one will ever run 75 kilometers per hour.
“In processes where there is such an intrinsic boundary, a kind of wall, you get a ‘thin’ tail: the slope decreases relatively quickly. You can also see this, for example, in the maximum age of people, which we conducted research into in 2019. People don’t live much longer than 115 years.
“Such a tail translates into what we call the extreme values index to call. That’s a number you can estimate. Not by just looking at the world records, which people often naively do. Then you only have a handful of data points.
“We have listed more than a thousand personal records for the 100 meters from top athletes. Out of that came one extreme values index which is negative. That means you have a thin tail, and therefore a physical boundary. We estimate that limit at approximately 9.36. Of course, that number also contains uncertainty, but in the near future no one will sprint much faster.”
And a fat tail?
“If you look at earthquakes, for example, geophysicists say: we actually know very little about how it works. There appears to be no inherent limit to the energy of earthquakes. Then we are talking about a fat tail, which you also see, for example, in the case of damage to fire or storm insurance.
“So suppose you have 100 billion in damage from an extreme storm or earthquake disaster, then the next one could just as easily cause 200 billion in damage. The frequency of more extreme events is decreasing very slowly.”
What do we, or insurers for example, benefit from such calculations?
“That is very useful for insurers, and important for risk analysis and premium calculations. Even if we could perfectly determine the probability distribution, there remains an uncertainty: we cannot change the natural variation, only quantify it.”
At first you feel like someone standing in the desert. Wherever you look around you, there is only sand
Your publications on sports records and the maximum age of people have received a lot of attention, but you are especially proud of your mathematical-statistical publications.
“These are my articles about methods to arrive at estimates and to determine the properties of those methods. It is very difficult to explain what you do there. It boils down to making certain assumptions and then trying to prove something mathematically based on them.
“In the beginning you feel like someone standing in the desert. Wherever you look around you, there is only sand. No idea where to go. But then slowly something starts to dawn, and then you see light at the end of the tunnel – that’s another metaphor.
“That may turn out to be a mirage, but you understand that you have to make different assumptions, and so you start again, and a landscape still emerges.
“I really like that game between assumptions and what you can prove based on them. Sometimes you hit a wall. And sometimes something suddenly occurs to you when you’re cycling home, or when you’re lying in bed at night.”
A silly question, but what do you think about the fact that this last name brought you into the theory of rare events?
“Ha, I’ve heard jokes about my name all my life. By the way, it’s not German: ‘einmal’ is without ‘h’. It is a Limburg surname that probably has to do with the town of Emael in Belgium, near Maastricht.
“I never realized the connection with my field, believe it or not. Until someone once made a name card: ‘Professor Einmahl, exception statistician’. Only then did it strike me.”