# Libres pensées d'un mathématicien ordinaire Posts

The Gumbel distribution appears in the modelling of extreme values phenomena and the famous Fisher–Tippett–Gnedenko theorem. Its cumulative distribution function is given by

$$x\in\mathbb{R}\mapsto\exp\left(-\exp\left(-\frac{x-\mu}{\sigma}\right)\right)$$

where $\mu\in\mathbb{R}$ and $\sigma\in(0,\infty)$ are location and scale parameters. Its mean and variance are

$$\mu+\gamma\sigma\quad\text{and}\quad\frac{\pi^2\sigma^2}{6},$$

where $\gamma=0.577\ldots$ is Euler’s constant. The maximum likelihood estimators $\hat{\mu}$ and $\hat{\sigma}$ satisfy

$$\hat{\mu} =-\hat{\sigma}\log\left(\frac{1}{n}\sum_{i=1}^{n}\mathrm{e}^{-x_i/\hat{\sigma}}\right)\quad\text{and}\quad\hat{\sigma} = \frac{\sum_{i=1}^nx_i}{n}-\frac{\sum_{i=1}^{n} x_i\mathrm{e}^{-x_i/\hat{\sigma}}}{ \sum_{i=1}^{n} \mathrm{e}^{-x_i/\hat{\sigma}}}.$$

Note that the second formula is implicit. Unfortunately, the current version of the Julia package Distributions.jl does not implement yet the Gumbel distro in fit()/fit_mle(). Here is a home brewed program to fit a Gumbel distribution on data using maximum likelihood and moments. The produced graphic is above. Note that the GNU R software implements a Gumbel fit in the fitdistrplus package with a notable contribution from my colleague Christophe Dutang.

# Pkg.add("Roots") # https://github.com/JuliaMath/Roots.jl
using Roots
"""
Computes approximately the Maximum Likelihood Estimator (MLE) of
the position and scale parameters of a Gumbel distribution with
cumulative distribution function x-> exp(-exp(-(x-mu)/sigma)).
Has mean mu+eulergamma*sigma and variance sigma^2*pi^2/6.
The MLE for mu is a function of data and the MLE for sigma.
The MLE for sigma is the root of a nonlinear function of data,
computed numerically with function fzero() from package Roots.

Reference: page 24 of book ISBN 1-86094-224-5.
Extreme value distributions.Theory and applications.
Imperial College Press, London, 2000. viii+187 pp.
https://ams.org/mathscinet-getitem?mr=1892574

no distribution should be stated without an explanation of how the
parameters are estimated even at the risk that the methods used will
not stand up to the present rigorous requirements of mathematically
minded statisticians''. E. J. Gumbel (1958).
"""
function gumbel_fit(data)
f(x) = x - mean(data) + sum(data.*exp(-data/x)) / sum(exp(-data/x))
sigma = fzero(f,sqrt(6)*std(data)/pi) # moment estimator as initial value
mu = - sigma * log(sum(exp(-data/sigma))/length(data))
return mu , sigma
end #function

# Pkg.add("Plots") # https://github.com/JuliaPlots/Plots.jl
# http://docs.juliaplots.org/
using Plots
function gumbel_fit_plots(data,label,filename)
n = length(data)
data = reshape(data,n,1)
mu , sigma = gumbel_fit(data)
X = collect(linspace(minimum(data),maximum(data),1000))
tmp = (X-mu)/sigma
Y = exp(-tmp-exp(-tmp))/sigma
pyplot()
histogram!(data,
nbins = round(Int,sqrt(n)),
normed = true,
label = @sprintf("%s histogram n=%i",label,n),
color = :white)
plot!(X,Y,
title = @sprintf("Gumbel density mu=%d sigma=%d",mu,sigma),
titlefont = font("Times", 10),
label = "Gumbel density",
lw = 3,
linestyle = :solid,
linecolor = :darkblue,
grid = false,
border = false)
gui()
#savefig(string(filename,".svg"))
savefig(string(filename,".png"))
end #function

function gumbel_fit_example()
gumbel_fit_plots(-log(-log(rand(1,1000))),"Sample","gumbelfitexample")
end #function

This short post is devoted to a couple of Julia programs.

Gaussian Unitary Ensemble. It is the Boltzmann-Gibbs measure with density proportional to
$$(x_1,\ldots,x_N)\in\mathbb{R}^N\mapsto\mathrm{e}^{-N\sum_{i=1}^N x_i^2}\prod_{i<j}(x_i-x_j)^2.$$
It is the distribution of the eigenvalues of a random $N\times N$ Hermitian matrix distributed according to the Gaussian probability measure with density proportional to
$$H\mapsto\mathrm{e}^{-N\mathrm{Tr}(H^2)}.$$
It is a famous exactly solvable model of mathematical physics. There is a nice formula for the mean empirical spectral distribution $\mathbb{E}\mu_N$ where $\mu_N=\frac{1}{N}\sum_{i=1}^N\delta_{x_i}$, namely
$$x\in\mathrm{R}\mapsto\frac{\mathrm{e}^{-\frac{N}{2}x^2}}{\sqrt{2\pi N}}\sum_{\ell=0}^{N-1}P_\ell^2(\sqrt{N}x)$$
where ${(P_\ell)}_{\ell\geq0}$ are the Hermite polynomials which are the orthonormal polynomials for the standard Gaussian distribution $\mathcal{N}(0,1)$. The computation of the Laplace transform and a subtle reasonning, see this paper, reveal that it converges as $N\to\infty$ towards the the Wigner semicircle distribution with density with respect to the Lebesgue measure
$$x\in\mathbb{R}\mapsto\frac{\sqrt{4-x^2}}{2\pi}\mathbf{1}_{x\in[-2,2]}.$$
Here is a nice plot followed by the Julia code used to produce it.

function normalized_hermite_polynomials_numeric(n,x)
if size(x,2) != 1
error("Second argument must be column vector.")
elseif n == 0
return ones(length(x),1)
elseif n == 1
return x
else
P = [ ones(length(x),1) x ]
for k = 1:n-1
P = [ P  x.*P[:,k+1]/sqrt(k+1)-P[:,k]*sqrt(1/(1+1/k)) ]
end
return P # matrix with columns P_0(x),...,P_n(x)
end
end #function

function gue_numeric(n,x)
if size(x,2) != 1
error("Second argument must be column vector.")
else
y = sqrt(n)*x
p = normalized_hermite_polynomials_numeric(n-1,y).^2
return exp(-y.^2/2) .* sum(p,2) / sqrt(2*pi*n)
end
end #function

using Plots
pyplot()
N = 8
X = collect(-3:.01:3)
Y = gue_numeric(N,X)
function semicircle(x)
if (abs(x)>2) return 0 else return sqrt(4-x^2)/(2*pi) end
end # function
Y = [Y [semicircle(x) for x in X]]
plot(X,Y,
title = @sprintf("GUE mean Empirical Spectral Distribution beta=2 N=%i",N),
titlefont = font("Times", 10),
label = ["GUE mean ESD" "SemiCircle"],
lw = 2,
linestyle = [:dash :solid],
linecolor = [:darkblue :darkred],
aspect_ratio = 2*pi,
grid = false,
border = false)
gui()
#savefig("gue.svg")
savefig("gue.png")


Here is, just for fun, a way to produce symbolically Hermite polynomials.

# Pkg.add("Polynomials") # http://juliamath.github.io/Polynomials.jl/latest/
using Polynomials

function unnormalized_hermite_polynomial_symbolic(n)
if n == 0
return Poly([1])
elseif n == 1
return Poly([0,1])
else
P = [Poly([1]) Poly([0,1])]
for k = 1:n-1
P = [ P Poly([0,1])*P[k+1]-k*P[k] ]
end
return P # better than just P[n+1]
end
end #function


Complex Ginibre Ensemble. It is the Boltzmann-Gibbs measure with density proportional to
$$(x_1,\ldots,x_N)\in\mathbb{C}^N\mapsto\mathrm{e}^{-N\sum_{i=1}^N|x_i|^2}\prod_{i<j}|x_i-x_j|^2.$$
It is the distribution of the eigenvalues of a random $N\times N$ complex matrix distributed according to the Gaussian probability measure with density proportional to
$$M\mapsto\mathrm{e}^{-N\mathrm{Tr}(MM^*)}$$
where $M^*=\overline{M}^\top$. Yet another exactly solvable model of mathematical physics, with a nice formula for the mean empirical spectral distribution $\mathbb{E}\mu_N$ where $\mu_N=\frac{1}{N}\sum_{i=1}^N\delta_{x_i}$, given by
$$z\in\mathbb{C}\mapsto\frac{\mathrm{e}^{-N|z|^2}}{\pi}\sum_{\ell=0}^{N-1}\frac{|\sqrt{N}z|^{2\ell}}{\ell!}$$
which is the analogue of the one given above for the Gaussian Unitary Ensemble. Using induction and integration by parts, it turns out that this density can be rewritten as
$$z\in\mathbb{C}\mapsto\int_{N|z|^2}^\infty\frac{u^{N-1}\mathrm{e}^{-u}}{\pi(N-1)!}\mathrm{d}u =\frac{\Gamma(N,N|z|^2)}{\pi}$$
where $\Gamma$ is the normalized incomplete Gamma function and where we used the identity
$$\mathrm{e}^{-r}\sum_{\ell=0}^n\frac{r^\ell}{\ell!}=\frac{1}{n!}\int_r^\infty u^n\mathrm{e}^{-u}\mathrm{d}u.$$
Note that the function $t\mapsto 1-\Gamma(N,t)$ is the cumulative distribution function of the Gamma distribution with shape parameter $N$ and scale parameter $1$. As a consequence, denoting $X_1,\ldots,X_N$ i.i.d. exponential random variables of unit mean, the density can be written as
$$z\in\mathbb{C}\mapsto\frac{1}{\pi}\mathbb{P}\left(\frac{X_1+\cdots+X_N}{N}\geq|z|^2\right).$$
Now the law of large numbers implies that this density converges pointwise as $N\to\infty$ towards the density of the uniform distribution on the unit disk of the complex plane
$$z\in\mathbb{C}\mapsto\frac{\mathbf{1}_{|z|\leq1}}{\pi}.$$
One can also use i.i.d. Poisson random variables $Y_1,\ldots,Y_N$ of mean $|z|^2$, giving
$$z\in\mathbb{C}\mapsto\frac{1}{\pi}\mathbb{P}\left(\frac{Y_1+\cdots+Y_N}{N}<1\right),$$
see this former post. Here are the plots with respect to $|z|$, followed by the Julia code.

# Pkg.add("Distributions") # https://juliastats.github.io/Distributions.jl
using Plots
using Distributions
N = 8
X = collect(0:.01:3)
Y = ccdf(Gamma(N,1),N*X.^2)/pi
function unit(x)
if (x>1) return 0 else return 1/pi end
end # function
Y = [Y [unit(x) for x in X]]
pyplot()
plot(X,Y,
title = @sprintf("Ginibre mean Empirical Spectral Distribution beta=2 N=%i",N),
titlefont = font("Times", 10),
label = ["Ginibre mean ESD" "Equilibrium"],
lw = 2,
linestyle = [:dash :solid],
linecolor = [:darkblue :darkred],
grid = false,
border = false)
gui()
#savefig("ginibre.svg")
savefig("ginibre.png")

Further reading. The Introducing Julia wikibook on Wikibooks.

This post presents an inequality for self-normalized sums without moment assumptions.

Symmetric laws. Recall that a probability distribution is symmetric when ${X}$ and ${-X}$ are equally distributed if ${X}$ is a random variable following this distribution. In this case ${\varepsilon=\mathrm{sign}(X)}$ and ${|X|}$ are independent and ${\varepsilon}$ follows a symmetric Rademacher distribution: ${\mathbb{P}(\varepsilon=\pm1)=1/2}$.

Concentration. Let ${X_1,\ldots,X_n}$ be independent real random variables with symmetric law and without atom at ${0}$. Then for any real ${r>0}$,

$\mathbb{P}\left(\frac{X_1+\cdots+X_n}{\sqrt{X_1^2+\cdots+X_n^2}}\geq r\right) \leq\mathrm{e}^{-\frac{r^2}{2}}.$

Note that this is available without any moment assumption on the random variables.

Proof. Thanks to the independence and symmetry assumptions, the random variables ${\varepsilon_1=\mathrm{sign}(X_1),\ldots,\varepsilon_n=\mathrm{sign}(X_n)}$ are iid, follow the symmetric Rademacher distribution, and are independent of ${|X_1|,\ldots,|X_n|}$. Now by conditioning we get

$\mathbb{P}\left(\frac{X_1+\cdots+X_n}{\sqrt{X_1^2+\cdots+X_n^2}}\geq r\right) =\mathbb{E}(\varphi_r(|X_1|,\ldots,|X_n|))$

where ${\varphi_r(c_1,\ldots,c_n)=\mathbb{P}((\varepsilon_1c_1+\cdots+\varepsilon_nc_n)/\sqrt{c_1^2+\cdots+c_n^2}\geq r)}$. We can assume that ${c_i>0}$ since ${\mathbb{P}(X_i=0)=0}$. It remains to use the Hoeffding inequality, which states that if ${Z_1,\ldots,Z_n}$ are independent centered and bounded real random variables then for any real ${r>0}$,

$\mathbb{P}\left(Z_1+\cdots+Z_n\geq r\right) \leq\exp\left(-\frac{2r^2}{\mathrm{osc}(Z_1)^2+\cdots+\mathrm{osc}(Z_n)^2}\right).$

where ${\mathrm{osc}(Z)=\max(Z)-\min(Z)}$. Here we use it with, for any ${i=1,\ldots,n}$,

$Z_i=\frac{c_i}{\sqrt{c_1^2+\cdots+c_n^2}}\varepsilon_i \quad\text{for which}\quad \mathrm{osc}(Z_i)^2=\frac{4c_i^2}{c_1^2+\cdots+c_n^2}.$

Indeed this gives ${ \varphi_r(c_1,\ldots,c_n) =\mathbb{P}\left(Z_1+\cdots+Z_n\geq r\right) \leq\mathrm{e}^{-\frac{r^2}{2}}}$.

Probabilistic interpretation. When ${X_1,\ldots,X_n}$ are iid and in ${L^2}$, then their mean is zero, and their variance is say ${\sigma^2>0}$. The law of large numbers gives ${\sqrt{X_1^2+\cdots+X_n^2}=\sqrt{n}(\sigma+o_{n\rightarrow\infty}(1))}$ almost surely. Therefore by the central limit theorem and Slutsky’s lemma we get ${(X_1+\cdots+X_n)/\sqrt{X_1^2+\cdots+X_n^2}\overset{\text{law}}{\longrightarrow}\mathcal{N}(0,1)}$ as ${n\rightarrow\infty}$.

Geometric interpretation. If ${X_1,\ldots,X_n}$ are iid standard Gaussian, then

$\frac{X_1+\cdots+X_n}{\sqrt{X_1^2+\cdots+X_n^2}} =\langle U_n,\theta_n\rangle$

where

$U_n=\sqrt{n}\frac{(X_1,\ldots,X_n)}{\sqrt{X_1^2+\cdots+X_n^2}} \quad\text{and}\quad \theta_n=\frac{(1,\ldots,1)}{\sqrt{n}}.$

The random vector ${U_n}$ is uniformly distributed on the sphere of ${\mathbb{R}^n}$ of radius ${\sqrt{n}}$, while the vector ${\theta_n}$ belongs to the unit sphere. Note that ${\langle U_n,\theta_n\rangle}$ is the law of the sum of the coordinates of a row or column of a uniform random orthogonal matrix.

Relation to Studentization. The result above can be related to the Studentized version of the empirical mean. Indeed, if one defined the empirical mean and the empirical variance

$\overline{X}_n=\frac{X_1+\cdots+X_n}{n} \quad\text{and}\quad \widehat\sigma^2_n=\frac{(X_1-\overline{X}_n)^2+\cdots+(X_n-\overline{X}_n)^2}{n-1}$

then using

$(n-1)\widehat\sigma_n^2 =X_1^2+\cdots+X_n^2-\frac{(X_1+\cdots+X_n)^2}{n}$

we get, for any ${r\geq0}$, after some algebra,

$\left\{\sqrt{n}\frac{\overline{X}_n}{\widehat\sigma_n}\geq r\right\} =\left\{\frac{X_1+\cdots+X_n}{\sqrt{X_1^2+\cdots+X_n^2}} \geq r\sqrt{\frac{n}{n-1+r^2}}\right\}.$

It follows then from the concentration inequality above that if ${X_1,\ldots,X_n}$ are independent, with symmetric law without atom at ${0}$, then for any ${r\geq0}$,

$\mathbb{P}\left(\sqrt{n}\frac{\overline{X}_n}{\widehat\sigma_n}\geq r\right) \leq\exp\left(-\frac{nr^2}{2(n-1+r^2)}\right).$

If ${X_1,\ldots,X_n}$ are iid centered Gaussian then ${\overline{X}_n\sim\mathcal{N}(0,1)}$ and ${\widehat{\sigma}^2_n\sim\chi^2(n-1)}$ are independent and their ratio ${\sqrt{n}\overline{X}_n/\widehat\sigma_n}$ follows the Student ${t(n-1)}$ law, of density proportional to ${x\mapsto 1/(1+t^2/(n-1))^{n/2}}$, which is in particular heavy tailed.

Extension and further reading. On this subject, one may take a look for instance at the article by Sergey Bobkov and Friedrich Götze entitled Concentration inequalities and limit theorems for randomized sums (2007). In another direction, it is also possible to show that a Cramér large deviation principle for self-normalized sums is available without any moment assumption, and we refer for instance to the probability survey by Qi-Man Shao and Qiying Wang entitled Self-normalized limit theorems: a survey (2013).

Many thanks to the mysterious L.

Sean O’Rourke pointed out on December 30, 2017 that a notation should be corrected in the statement of Lemma A.1 in the probability survey Around the circular law (2012) that I wrote years ago in collaboration with Charles Bordenave.

Indeed the definition of ${\sigma^2}$ should be corrected to

$\sigma^2 :=\min_{1\leq i,j\leq n}\mathrm{Var}(X_{ij}\mid|X_{i,j}|\leq a)>0.$

It was erroneously written

$\sigma^2 :=\min_{1\leq i,j\leq n}\mathrm{Var}(X_{ij}\mathbf{1}_{|X_{i,j}|\leq a})>0.$

Let us take this occasion for a back to basics about conditional variance and variance of truncation. Let ${X}$ be a real random variable on ${(\Omega,\mathcal{F},\mathbb{P})}$ and ${A\in\mathcal{F}}$ be an event. First the real number ${\mathbb{E}(X\mid A)=\mathbb{E}(X\mid\mathbf{1}_A=1)}$ is not the random variable ${\mathbb{E}(X\mid\mathbf{1}_A)}$. We have

$\mathbb{E}(X\mid\mathbf{1}_A) =\underbrace{\frac{\mathbb{E}(X\mathbf{1}_A)}{\mathbb{P}(A)}}_{\mathbb{E}(X\mid A)}\mathbf{1}_A +\underbrace{\frac{\mathbb{E}(X\mathbf{1}_{A^c})}{\mathbb{P}(A^c)}}_{\mathbb{E}(X\mid A^c)}\mathbf{1}_{A^c}.$

Note that this formula still makes sense when ${\mathbb{P}(A)=0}$ or ${\mathbb{P}(A)=1}$.

The quantity ${\mathbb{E}(X\mid A)}$ makes sense only if ${\mathbb{P}(A)>0}$, and in this case, the conditional variance of ${X}$ given the event ${A}$ is the real number given by

$\begin{array}{rcl} \mathrm{Var}(X\mid A) &=&\mathbb{E}((X-\mathbb{E}(X\mid A))^2\mid A)\\ &=&\mathbb{E}(X^2\mid A)-\mathbb{E}(X\mid A)^2\\ &=&\frac{\mathbb{E}(X^2\mathbf{1}_A)}{\mathbb{P}(A)} -\frac{\mathbb{E}(X\mathbf{1}_A)^2}{\mathbb{P}(A)^2}\\ &=& \frac{\mathbb{E}(X^2\mathbf{1}_A)\mathbb{P}(A)-\mathbb{E}(X\mathbf{1}_A)^2}{\mathbb{P}(A)^2}\\ &=&\mathbb{E}_A(X^2)-\mathbb{E}_A(X)^2=:\mathrm{Var}_A(X) \end{array}$

where ${\mathbb{E}_A}$ is the expectation with respect to the probability measure with density ${\mathbf{1}_A/\mathbb{P}(A)}$ with respect to ${\mathbb{P}}$. In particular, by the Cauchy–Schwarz inequality,

$\mathrm{Var}(X\mid A) \geq 0$

with equality if and only if ${X}$ and ${\mathbf{1}_A}$ are colinear.

Of course ${\mathrm{Var}(X\mid A)=0}$ if ${X}$ is constant. However ${\mathrm{Var}(X\mid A)}$ may vanish for a non-constant ${X}$. Indeed if ${A=\{|X|\leq a\}}$ and if ${X\sim\frac{1}{2}\delta_{a/2}+\frac{1}{2}\delta_{2a}}$ then ${X\mid A}$ is constant and equal to ${a/2}$. In this example, since ${X\mathbf{1}_A}$ is not a constant, this shows also that one cannot lower bound ${\mathrm{Var}(X\mid A)}$ with the variance of the truncation

$\mathrm{Var}(X\mathbf{1}_A)=\mathbb{E}(X^2\mathbf{1}_A)-\mathbb{E}(X\mathbf{1}_A)^2.$

Another notable correction. Mylène Maïda pointed out to me on February 27 2018 that at the bottom of page 14, just before the statement

$\sup_{z\in C}|n\varphi_{n,1}(\sqrt{n}z)-\pi^{-1}\mathbf{1}_{[0,1]}(|z|)|=0$

the compact set ${C}$ must be taken in ${\{z\in\mathbb{C}:|z|\neq1\}}$ and not on the whole complex plane ${\mathbb{C}}$. Indeed, when ${|z|=1}$, ${n\varphi_{n,1}(\sqrt{n}z)}$ tends as ${n\rightarrow\infty}$ to ${1/2}$, and not to ${\pi^{-1}}$, see for instance this former post for a one formula proof based on the central limit theorem for Poisson random variables. Anyway this is really not surprising since a sequence of continuous functions cannot converge uniformly to a discontinuous function.