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Éléments de statistiquePartie I - Statistique fondamentale
Etienne Roquain(merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...)
Master 2 Probabilités et Finance
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) 1 / 23
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Chapitre 1. Modèle statistique
1 Données
2 Modèle statistique
3 Trois grands problèmes statistiques
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) 2 / 23
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1 Données
2 Modèle statistique
3 Trois grands problèmes statistiques
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 3 / 23
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Tirages pile ou face
0 200 400 600 800 1000
0.0
0.4
0.8
I Pièce équilibrée?I Nombre de tirages nécessaires?
Source : simulations en R
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 4 / 23
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Caractéristiques corporelles
Taille Poids Tour de cheville
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0 100 200 300 400 500
1618
2022
2426
28
I Taille des femmes suit une loi gaussienne?I Prédiction poids à partir du tour de cheville?
Source : https ://ww2.amstat.org/publications/jse/v11n2/datasets.heinz.html
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 5 / 23
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Caractéristiques appareils photos1000 appareils photos, 13 caractéristiques
I Prédiction prix en fonction des autres caractéristiques?I Prix lié à la dimension de l’appareil ?
Source : https ://www.kaggle.com/crawford/1000-cameras-datasetEtienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 6 / 23
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Données qualitatives
Cheveux\Yeux Marron Bleu Noisette orangeBrun 68 20 15 5
Châtain 119 84 54 29Roux 26 17 14 14Blond 7 94 10 16
Yeux\Sexe Homme FemmeMarron 98 122
Bleu 101 114Noisette 47 46orange 33 31
I Couleur des yeux liée à la couleur des cheveux?
Source : R-package HairEyeColor
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 7 / 23
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Mesures de polluants par laser
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400 500 600 700
−0.
8−
0.6
−0.
4−
0.2
0.0
I Courbe physique sous-jacente?
Source : R-package SemiParEtienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 8 / 23
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Expression de gènes
gènesm
alad
esa
in
I Effets des gènes sur le cancer de la prostate?I Gènes impliqués dans le cancer de la prostate?
Source : R-package sda
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 9 / 23
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Reconstitution de signal dimension 1
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0.0 0.2 0.4 0.6 0.8 1.0
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0.0 0.2 0.4 0.6 0.8 1.0
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−1
01
I Débruitage?
Source : simulations en R
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 10 / 23
-
Reconstitution de signal dimension 2
I Débruitage?
Source : R-package EMD et simulations en R
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 11 / 23
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1 Données
2 Modèle statistique
3 Trois grands problèmes statistiques
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 12 / 23
-
Probabilité vs Statistique
I Soit (X1, . . . ,Xn) : (Ω,F)→ (X n,Xn) mesurable
I Probabilité : loi Pn de (X1, . . . ,Xn) donnée
(Ω,F ,P) −→ (X n,Xn,Pn)ω 7−→ (X1(ω), . . . ,Xn(ω)) = (x1, . . . , xn)
I Statistique : réalisation (x1, . . . , xn) ∈ X n , loi Pn ?
Modèle Pn ∈ P = {Pn,θ, θ ∈ Θ} lois candidates
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 13 / 23
-
Probabilité vs Statistique
I Soit (X1, . . . ,Xn) : (Ω,F)→ (X n,Xn) mesurable
I Probabilité : loi Pn de (X1, . . . ,Xn) donnée
(Ω,F ,P) −→ (X n,Xn,Pn)ω 7−→ (X1(ω), . . . ,Xn(ω)) = (x1, . . . , xn)
I Statistique : réalisation (x1, . . . , xn) ∈ X n , loi Pn ?
Modèle Pn ∈ P = {Pn,θ, θ ∈ Θ} lois candidates
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 13 / 23
-
Probabilité vs Statistique
I Soit (X1, . . . ,Xn) : (Ω,F)→ (X n,Xn) mesurable
I Probabilité : loi Pn de (X1, . . . ,Xn) donnée
(Ω,F ,P) −→ (X n,Xn,Pn)ω 7−→ (X1(ω), . . . ,Xn(ω)) = (x1, . . . , xn)
I Statistique : réalisation (x1, . . . , xn) ∈ X n , loi Pn ?
Modèle Pn ∈ P = {Pn,θ, θ ∈ Θ} lois candidates
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 13 / 23
-
Probabilité vs Statistique
I Soit (X1, . . . ,Xn) : (Ω,F)→ (X n,Xn) mesurable
I Probabilité : loi Pn de (X1, . . . ,Xn) donnée
(Ω,F ,P) −→ (X n,Xn,Pn)ω 7−→ (X1(ω), . . . ,Xn(ω)) = (x1, . . . , xn)
I Statistique : réalisation (x1, . . . , xn) ∈ X n , loi Pn ?
Modèle Pn ∈ P = {Pn,θ, θ ∈ Θ} lois candidates
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 13 / 23
-
Cadre formel
DéfinitionUne expérience statistique est définie par un triplet (X n,Xn,P) où (X ,X) estun espace mesurable et P est une famille de lois sur (X n,Xn).
Vocabulaire :I P est le modèle statistique.I L’entier n ≥ 1 est la taille de l’échantillon.I L’espace (X ,X) (ou (X n,Xn)) est l’espace d’observation (X = R où Rp).I Une paramétrisation du modèle P est une application θ ∈ Θ 7→ Pn,θ
satisfaisant P = {Pn,θ, θ ∈ Θ}.I L’ensemble Θ est l’espace des paramètres et θ est le paramètre du
modèle.Une observation (X1, . . . ,Xn) du modèle satisfait :
(Ω,F , {Pθ, θ ∈ Θ}) −→ (X n,Xn, {Pn,θ, θ ∈ Θ})ω 7−→ (X1(ω), . . . ,Xn(ω)) = (x1, . . . , xn)
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 14 / 23
-
Types de modèlesLe modèle P = {Pn,θ, θ ∈ Θ} (ainsi paramétré) est dit . . .
. . . identifiablesi ∀θ, θ′ ∈ Θ, Pn,θ = Pn,θ′ implique θ = θ′.
Sinon reconstruction du paramètre impossible à l’aide des données !
. . . produit
s’il existe {Pθ, θ ∈ Θ}, famille de lois sur (X ,X), tell que Pn,θ = P⊗nθ , θ ∈ Θ(c’est-à-dire X1, . . . ,Xn i.i.d.)
Convention : pour les modèles produits, la famille {Pθ, θ ∈ Θ} est le modèle.
. . . paramétrique
s’il existe p ≥ 1 tel que Θ ⊂ Rp, il est dit non-paramétrique sinon
. . . de grande dimension
si Θ ⊂ Rp, Vect(Θ) = Rp avec p « grand » (informel)Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 15 / 23
-
Tirages pile ou face
0 200 400 600 800 1000
0.0
0.4
0.8
I Modèle {B(θ)⊗n, θ ∈ [0,1]}I Identifiable, produit, paramétrique.
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 16 / 23
-
Tirages pile ou face
0 200 400 600 800 1000
0.0
0.4
0.8
I Modèle {B(θ)⊗n, θ ∈ [0,1]}I Identifiable, produit, paramétrique.
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 16 / 23
-
Caractéristiques corporelles
Taille Poids Tour de cheville
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1618
2022
2426
28
I Par exemple modèle {N (µ, σ2)⊗n, θ = (µ, σ) ∈ R× (0,∞)}I Identifiable, produit, paramétriqueI Mais modèle {N (µ, σ2)⊗n, θ = (µ, σ) ∈ R2} pas identifiable.
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 17 / 23
-
Caractéristiques corporelles
Taille Poids Tour de cheville
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1618
2022
2426
28
I Par exemple modèle {N (µ, σ2)⊗n, θ = (µ, σ) ∈ R× (0,∞)}I Identifiable, produit, paramétriqueI Mais modèle {N (µ, σ2)⊗n, θ = (µ, σ) ∈ R2} pas identifiable.
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 17 / 23
-
Caractéristiques corporelles
Taille Poids Tour de cheville
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0 100 200 300 400 500
1618
2022
2426
28
I Par exemple modèle {N (µ, σ2)⊗n, θ = (µ, σ) ∈ R× (0,∞)}I Identifiable, produit, paramétriqueI Mais modèle {N (µ, σ2)⊗n, θ = (µ, σ) ∈ R2} pas identifiable.
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 17 / 23
-
Caractéristiques corporelles
Taille Poids Tour de cheville
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1618
2022
2426
28
I Autre exemple modèle P = {P⊗n, P loi sur R }, θ = PI Identifiable, produit, non-paramétrique
Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 18 / 23
-
Caractéristiques corporelles
Taille Poids Tour de cheville
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�