Chapitre 8: Distribution d’un estima- teur
1. Distribution de la moyenne arithm´etique
2. Bootstrap
Au chapitre pr´ec´edent, nous avons consid´er´e le probl`eme de l’estimation de caract´e- ristiques de la distribution d’une variable, comme sa moyenne ou sa variance. Nous avons d´efini un estimateur comme une fonction des observations dont on se sert pour estimer ces caract´eristiques. Nous avons remarqu´e qu’un estimateur est lui-mˆeme une variable al´eatoire. La pr´ecision des estimations fournies par un estimateur va d´ependre des caract´eristiques de sa distribution.
Consid´erons un ´echantillon d’observations x 1 , ..., x n issues de variables al´eatoires X 1 , ..., X n i.i.d. ∼ F X . Un estimateur est une fonction B(X 1 , ..., X n ). Sa valeur observ´ee sur l’´echantillon est b = B (x 1 , ..., x n ) et on l’appelle une estimation.
On voit bien que l’estimation serait diff´erente si l’´echantillon ´etait diff´erent. On peut donc
d´efinir la distribution de B sur la population de tous les ´echantillons de taille n, appel´ee
distribution d’´echantillonnage et not´ee F B . Evidemment, F B va d´ependre de F X qui
n’est pas connue dans la pratique. Pour l’approcher, on pourra utiliser soit un mod`ele
math´ematique d´epandant de param`etres (approche param´etrique), soit la fonction de
distribution cumulative empirique F n (approche non param´etrique).
1. Distribution de la moyenne arithm´etique
Nous avons vu au chapitre pr´ec´edent que la moyenne arithm´etique X = 1
n
n X i=1
X i
est l’estimateur du maximum de vraisemblance de l’esp´erance math´ematique pour de nombreux mod`eles de distributions:
• Distribution normale: µ ˆ = X pour X 1 , ..., X n i.i.d. ∼ N (µ, σ 2 )
• Distribution de Poisson: ˆ λ = X pour X 1 , ..., X n i.i.d. ∼ P (λ)
• Distribution binomiale: p ˆ = X pour X 1 , ..., X n i.i.d. ∼ B(1, p)
Nous allons nous int´eresser aux propri´et´es de X de fa¸con g´en´erale.
Soient X 1 , ..., X n i.i.d. ∼ F X avec E(X i ) = µ et var(X i ) = σ 2 , i = 1, ..., n.
• Esp´erance de X : en applicant les propri´et´es de l’esp´erance, on trouve E (X ) = E
1 n
n X i=1
X i
= 1 n E
n X i=1
X i
= 1 n
n X i=1
E (X i ) = 1
n nµ = µ.
L’esp´erance math´ematique de l’estimateur X est donc ´egale ` a l’esp´erance math´e- matique des X i . Cela signifie qu’en moyenne, l’estimation x fournie par X sur un
´echantillon vaudra E(X i ), qui est pr´ecis´ement la carat´eristique que nous voulions estimer. On dit que X est un estimateur sans biais de E (X i ).
• Variance de X : en applicant les propri´et´es de la variance et en utilisant l’ind´ependance des X i , on trouve
var(X ) = var
1 n
n X i=1
X i
= 1 n 2 var
n X i=1
X i
= 1 n 2
n X i=1
var(X i ) = 1
n 2 nσ 2 = σ 2 n . La variance de la moyenne arithm´etique est ´egale ` a la variance des X i divis´ee par la taille de l’´echantillon. La pr´ecision de l’estimation augmente donc avec la taille de l’´echantillon.
• Ecart-type de X : le r´esultat pour la variance implique sd(X ) = √ σ .
Quelle est la distribution de X ?
→ Loi normale:
Propri´et´e d’additivit´e de la loi normale: Soient X 1 ∼ N (µ 1 , σ 1 2 ) et X 2 ∼ N (µ 2 , σ 2 2 ) ind´ependantes. Alors
(X 1 + X 2 ) ∼ N (µ 1 + µ 2 , σ 1 2 + σ 2 2 ).
En utilisant cela, on obtient X =
1 n
n X i=1
X i
∼ N µ, σ 2 n
!
si X 1 , ..., X n i.i.d. ∼ N (µ, σ 2 ).
Pour les autres distributions, le r´esultat ci-dessus reste vrai approximativement et pour les grands ´echantillons grˆ ace au r´esultat fondamental suivant:
Th´ eor` eme central limite
Soient X 1 , ..., X n i.i.d. ∼ F X avec E (X i ) = µ et var(X i ) = σ 2 , i = 1, ..., n, soit X = 1
n
P n
i=1 X i et soit
V = X − µ σ/ √
n ∼ F V .
V est la moyenne arithm´etique centr´ ee et r´ eduite (on a soustrait ` a X son esp´erance et divis´e le r´esultat par son ´ecart-type). Alors
n→∞ lim F V (t) = Φ(t),
o` u Φ(t) est la cumulative de la distribution normale standard.
La cumulative d’une variable (de mˆeme que sa densit´e) d´etermine compl`etement sa distribution. Le r´esultat ci-dessus signifie donc que la moyenne arithm´etique centr´ ee et r´ eduite est approximativement normale N (0, 1) si n est suffisamment grand. Ceci implique que X est approximativement normale N
µ, σ n 2
.
Ce qui est remarquable, c’est que le r´esultat de la page pr´ec´edente est valable quelle que soit F X , la distribution des X i (pourvu que leur esp´erance et leur variance soient bien d´efinies).
Par contre, la taille d’´echantillon n ` a partir de laquelle l’approximation est bonne d´epend de F X , et il n’y a pas en g´en´eral de r`egle simple pour la d´eterminer.
Dans les pages qui suivent figurent trois exemples o` u on a repr´esent´e les histogrammes et les qq-plots de x pour diff´erents mod`eles F X et diff´erentes tailles d’´echantillon n.
Pour les obtenir, on a g´en´er´e ` a l’aide d’un ordinateur 1000 ´echantillons de taille n
d’observations suivant le mod`ele F X , et calcul´e ` a chaque fois la valeur de x.
X est uniforme entre 0 et 100.
Histogram of xbar n = 1
xbar
Density
0 20 60 100
0.000 0.004 0.008
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−3 −1 1 2 3
0 20 40 60 80 100
Normal Q−Q Plot
Theoretical Quantiles
Sample Quantiles
Histogram of xbar n = 5
xbar
Density
20 40 60 80
0.000 0.010 0.020 0.030
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−3 −1 1 2 3
20 40 60 80
Normal Q−Q Plot
Theoretical Quantiles
Sample Quantiles
Histogram of xbar n = 10
xbar
Density
20 40 60 80
0.00 0.01 0.02 0.03
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−3 −1 1 2 3
30 40 50 60 70 80
Normal Q−Q Plot
Theoretical Quantiles
Sample Quantiles
Histogram of xbar n = 15
xbar
Density
30 50 70
0.00 0.01 0.02 0.03 0.04 0.05
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−3 −1 1 2 3
30 40 50 60 70
Normal Q−Q Plot
Theoretical Quantiles
Sample Quantiles
Histogram of xbar n = 20
xbar
Density
30 50 70
0.00 0.01 0.02 0.03 0.04 0.05
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−3 −1 1 2 3
30 40 50 60 70
Normal Q−Q Plot
Theoretical Quantiles
Sample Quantiles
Histogram of xbar n = 25
xbar
Density
30 40 50 60 70
0.00 0.02 0.04 0.06
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