6 Estatísticas
Funções da amostra que não dependem de \(\theta \in \Theta\)
6.1 Exemplo
Seja \((X_{1}, \dots, X_{n})\) a.a. de \(X\sim f_{\theta,}\theta \in \Theta\). São estatísticas: 1. \(T_{1}(X_{1},\dots,X_{n})=X_{1}+\dots+X_{n}\) 2. \(T_{2}=\bar{X}= \frac{X_{1}+\dots+X_{n}}{n}\) 3. \(T_{3}=\max\{X_{1},\dots,X_{n}\}=X_{(n)}\) 4. \(T_{4}=\min\{X_{1},\dots,X_{n}\}=X_{(1)}\) 5. \(T_{5}(X_{1},\dots,X_{n})=X_{(n)}-X_{(1)}\) 6. \(T_{6}(X_{1},\dots,X_{n})=X_{i}, \text{para algum }i=1,\dots,n\) 7. \(T_{7}(X_{1},\dots,X_{n})=\frac{1}{n}\sum\limits^{n}_{i=1}(X_{i}-\bar{X})^{2}\) 8. \(T_{8}(X_{1},\dots,X_{n})=\frac{1}{n-1}\sum\limits^{n}_{i=1}(X_{i}-\bar{X})^{2}\) 9. \(T_{9}(X_{1},\dots,X_{n})=\frac{1}{n}\sum\limits^{n}_{i=1}|X_{i}-\bar{X}|\) 10. \(T_{10}(X_{1},\dots,X_{n})=\sqrt{\frac{1}{n}\sum\limits^{n}_{i=1}(X_{i}-\bar{X})^{2}}\) etc.
As estatísticas são variáveis aleatórias: \[ \begin{aligned} \pmb{X}_n&=(X_{1},\dots,X_{n}):\Omega\rightarrow \mathbb{R}^{n} \\ T(\pmb{X}_n)&=T \circ \pmb{X}_n:\Omega\rightarrow \mathbb{R} \end{aligned} \]