| Les deux révisions précédentesRévision précédenteProchaine révision | Révision précédente |
| math:2:problematique_estimation [2016/02/15 21:53] – Alain Guichet | math:2:problematique_estimation [2020/05/25 10:48] (Version actuelle) – Alain Guichet |
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| __**Exemples**__ | __**Exemples**__ |
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| - Le premier exemple ci-dessus correspond à la situation :\\ $$\ds\Theta=\left]0,1\right[\subset\R$$$$\mu_{\theta}=\mathcal{B}(1,\theta)$$$$g(\theta)=\theta$$$$\Omega_{\theta}=\left\{ 0,1\right\} ^{\N^{*}}$$$$\forall k\in\N^{*},\;\mathbb{P}_{\theta}(X_{k}=1)=\theta,\;\mathbb{P}_{\theta}(X_{k}=0)=1-\theta$$ | - Le premier exemple ci-dessus correspond à la situation : $$\ds\Theta=\left]0,1\right[\subset\R$$ $$\mu_{\theta}=\mathcal{B}(1,\theta)$$ $$g(\theta)=\theta$$ $$\Omega_{\theta}=\left\{ 0,1\right\} ^{\N^{*}}$$ $$\forall k\in\N^{*},\;\mathbb{P}_{\theta}(X_{k}=1)=\theta,\;\mathbb{P}_{\theta}(X_{k}=0)=1-\theta$$ |
| - **Moyenne empirique** :\\ $$\ds\varphi_{n}(x_{1},\dots,x_{n})=\frac{x_{1}+\dots+x_{n}}{n}$$$$\ds\bar{X}_{n}=\varphi_{n}(X_{1},\dots,X_{n})=\frac{1}{n}\left(X_{1}+\dots+X_{n}\right)$$ | - **Moyenne empirique** : $$\ds\varphi_{n}(x_{1},\dots,x_{n})=\frac{x_{1}+\dots+x_{n}}{n}$$ $$\ds\bar{X}_{n}=\varphi_{n}(X_{1},\dots,X_{n})=\frac{1}{n}\left(X_{1}+\dots+X_{n}\right)$$ |
| - **Variance empirique** :\\ $$\ds\varphi_{n}(x_{1},\dots,x_{n})=\frac{1}{n}\left(x_{1}^{2}+\dots+x_{n}^{2}\right)-\left[\frac{1}{n}\left(x_{1}+\dots+x_{n}\right)\right]^{2}$$$$\ds\bar{V}_{n}=\varphi_{n}(X_{1},\dots,X_{n})$$ | - **Variance empirique** : $$\ds\varphi_{n}(x_{1},\dots,x_{n})=\frac{1}{n}\left(x_{1}^{2}+\dots+x_{n}^{2}\right)-\left[\frac{1}{n}\left(x_{1}+\dots+x_{n}\right)\right]^{2}$$ $$\ds\bar{V}_{n}=\varphi_{n}(X_{1},\dots,X_{n})$$ |
| - Le second exemple ci-dessus correspond à la situation :\\ $$\Theta=\left\{ (a,b)\in\R^{2}\mid a<b\right\} \subset\R^{2}$$$$\mu_{(a,b)}=\mathcal{U}([a,b])$$$$g(a,b)=b-a$$$$\Omega_{(a,b)}=\left[a,b\right]^{\N^{*}}$$$$\ds\forall x\in\R,\forall k\in\N^{*},\;\mathbb{P}_{(a,b)}(X_{k}\leqslant x)=\frac{x-a}{b-a}1\!\!1_{[a,b]}(x)+1\!\!1_{[b,+\infty[}$$ | - Le second exemple ci-dessus correspond à la situation : $$\Theta=\left\{ (a,b)\in\R^{2}\mid a<b\right\} \subset\R^{2}$$ $$\mu_{(a,b)}=\mathcal{U}([a,b])$$ $$g(a,b)=b-a$$ $$\Omega_{(a,b)}=\left[a,b\right]^{\N^{*}}$$ $$\ds\forall x\in\R,\forall k\in\N^{*},\;\mathbb{P}_{(a,b)}(X_{k}\leqslant x)=\frac{x-a}{b-a}1\!\!1_{[a,b]}(x)+1\!\!1_{[b,+\infty[}$$ |
| - **Minimum empirique** :\\ $$\varphi_{n}(x_{1},\dots,x_{n})=\min\{x_{1},\dots,x_{n}\}$$$$\bar{I}_{n}=\varphi_{n}(X_{1},\dots,X_{n})=\min\{X_{1},\dots,X_{n}\}$$ | - **Minimum empirique** : $$\varphi_{n}(x_{1},\dots,x_{n})=\min\{x_{1},\dots,x_{n}\}$$ $$\bar{I}_{n}=\varphi_{n}(X_{1},\dots,X_{n})=\min\{X_{1},\dots,X_{n}\}$$ |
| - **Maximum empirique** :\\ $$\varphi_{n}(x_{1},\dots,x_{n})=\max\{x_{1},\dots,x_{n}\}$$$$\bar{S}_{n}=\varphi_{n}(X_{1},\dots,X_{n})=\max\{X_{1},\dots,X_{n}\}$$ | - **Maximum empirique** : $$\varphi_{n}(x_{1},\dots,x_{n})=\max\{x_{1},\dots,x_{n}\}$$ $$\bar{S}_{n}=\varphi_{n}(X_{1},\dots,X_{n})=\max\{X_{1},\dots,X_{n}\}$$ |
| - **Étendue empirique** :\\ $$\varphi_{n}(x_{1},\dots,x_{n})=\max\{x_{1},\dots,x_{n}\}-\min\{x_{1},\dots,x_{n}\}$$$$\bar{E}_{n}=\varphi_{n}(X_{1},\dots,X_{n})=\bar{S}_{n}-\bar{I}_{n}$$ | - **Étendue empirique** : $$\varphi_{n}(x_{1},\dots,x_{n})=\max\{x_{1},\dots,x_{n}\}-\min\{x_{1},\dots,x_{n}\}$$ $$\bar{E}_{n}=\varphi_{n}(X_{1},\dots,X_{n})=\bar{S}_{n}-\bar{I}_{n}$$ |
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