You find it cold in Montréal ? trust me, it is even worse

As people say in Montréal, “aujourd’hui, il fait frette”. And I have been surprised recently when some people told my that we would reach -35°C Sunday evening… I checked around, and I found -25°C on all weather forecast websites. But nowhere -35°C. I asked some friends, and they told me that those people were not really looking at the air temperature (as we observe on the thermometer), but they were looking at the wind chill, also called “felt air temperature on exposed skin due to the wind” (température ressentie).
And indeed, such a quantity does exist, and can be found on website. There is also a physical background for that quantity. Hence, the windchill is defined as

where is the air temperature (in °C), and the wind speed (in km/h). Please don’t ask me how to interpret this power 0.16 (I already find difficult to explain a square root in an econometric equation). If we look at the past previous days we observe the following observations,
where points on top are temperature, while below we have felt temperature.So, basically, winters are even colder than what you might think..

And the story is not over, yet. The same thing holds for summer: if you take into account humidity, summer are even hotter than what you think… There is thehumidex, defined here as

where denotes a dewpoint (see here for more details).

That index appeared in the 70’s, with a work of Masterson and Richardson entitled a method of quantifying human discomfort due to excessive heat and humidity (published in 1979).By that time, in Canada, on average, 22 people died, per year, because of those excessive heat and humidity. For those interested by the origin of that index, you can have a look here.
Recently, @Annmaria (here) told me that one might expect variance to increase, i.e. maximas should be increasing faster than minimas. I just wonder if this intuition can be related to the fact that more and more people (including some medias) now talk more about felt temperatures than measured temperatures. And if we compare past temperatures to felt temperature we have today, it looks like the difference between extremes is increasing….

Estimer une variance… ou un écart-type ?

Lors du cours de demain, nous allons parler (longuement) de l’estimation de la variance (comme cela avait commencé en TD vendredi dernier). Mais en pratique, c’est souvent l’écart-type qui nous intéresse. On va passer par mal de temps à chercher un estimateur sans biais de la variance, et montrer que

devrait convenir, mais cela ne nous donne pas vraiment d’information sur le biais de l’estimateur naturel de l’écart-type, à savoir

Bref, il y a sûrement un biais, et ça pourrait être intéressant de le quantifier. Classiquement, on suppose disposer d’un échantillon i.i.d. On va faire une hypothèse forte, à savoir que leur loi est une loi normale, où est inconnu.
Naturellement, on voudrait considérer

On va poser, afin de simplifier un peu,
et on va chercher à calculer

où est la densité de cette variable. Or on sait que suit une loi du chi-deux à degrés de liberté, dont la densité est la loi gamma suivante

En notant que l’on peut réécrire

on a que cette espérance s’écrit

soit en faisant un peu de calculs,

ou encore

Ce qui va se simplifier en utilisant la fonction gamma

On a ainsi