43  Regiões de confiança sob normalidade

43.1 Intervalos de confiança simultâneos para o vetor de médias (sob normalidade)

Seja \(\boldsymbol{X}_n^*\) a.a. de \(\boldsymbol{X} \sim N_p(\boldsymbol{\mu}, \Sigma)\). Uma região de confiança para \(g(\theta) = C \boldsymbol{\mu}\) com coeficiente de confiança \(\gamma \in (0,1)\) em que \(C_{s \times p}\) é uma matriz com linhas linearmnete independentes, pode ser definda utilizando a quantidade

\[ W(\theta) = \frac{n-s}{s} (C\bar{X} - C\boldsymbol{\mu})^T [C S^2_n C^T]^{-1} (C\bar{X} - C \boldsymbol{\mu}) \sim F_{(s, n-s)} \]

\[ \mathrm{RC}(g(\theta),\gamma) = \left\{g(\theta) : W(\theta) \leq q_{\gamma} \right\} \] em que \(g_{\gamma}\) é tal que \[ P(F_{(s,n-s)} \leq q_{\gamma}) = \gamma \]

Casos particulares:

  1. \[ \begin{aligned} g(\theta) &= \mu_1 \Rightarrow C\boldsymbol{\mu} = \begin{pmatrix} 1 & 0 & \dots & 0 \end{pmatrix} \begin{pmatrix} \mu_1 \\ \vdots \\ \mu_p \end{pmatrix} = \mu_1 \\ \Rightarrow W(\theta) &= \frac{n-1}{1} \frac{(\bar{X}_1 - \mu_1)^2}{S^2_{n,(1,1)}}\sim F_{(1, n-1)} \\ \Rightarrow \mathrm{RC}(\mu_1, \gamma) &= \left\{\mu_1 \in \mathbb{R} : \frac{(\bar{X}_1 - \mu_1)^2}{S^2_{n,(1,1)}} \leq q_{\gamma}\right\} \\ \Rightarrow \mathrm{RC}(\mu_1, \gamma) &= \bar{X}_1 \pm \sqrt{q_{\gamma} \frac{S^2_{n,(1,1)}}{n-1}} \end{aligned} \]

em que

\[ S^2_n = \frac{1}{n} \sum^n_{i=1} (X_i - \bar{X})(X_i - \bar{X})^T = \begin{bmatrix} S^2_{n,(1,1)} & \dots & S^2_{n,(1,p)} \\ \vdots & \ddots & \vdots \\ S^2_{n,(p,1)} & \dots & S^2_{n,(p,p)} \end{bmatrix} \]

e \[ P(F_{(1, n-1)} \leq q_{\gamma}) = \gamma. \]

  1. \[ \begin{aligned} g(\theta) &= \mu_j \Rightarrow C\boldsymbol{\mu} = \begin{pmatrix} 0 & 0 & \dots & 1 & 0 & \dots & 0 \end{pmatrix} \begin{pmatrix} \mu_1 \\ \vdots \\ \mu_p \end{pmatrix} = \mu_j \\ \Rightarrow W(\theta) &= \frac{n-1}{1} \frac{(\bar{X}_j - \mu_j)^2}{S^2_{n,(j,j)}} \sim F_{(1, n-1)} \\ \Rightarrow \mathrm{RC}(\mu_j, \gamma) &= \left\{\mu_1 \in \mathbb{R} : \frac{(\bar{X}_j - \mu_j)^2}{S^2_{n,(j,j)}} \leq q_{\gamma}\right\} \\ \Rightarrow \mathrm{RC}(\mu_j, \gamma) &= \bar{X}_j \pm \sqrt{q_{\gamma} \frac{S^2_{n,(j,j)}}{n-1}} \end{aligned} \]

em que \(p > j\)

  1. \[ \begin{aligned} g(\theta) &= \mu_1 - \mu_2 + 3\mu_3 \Rightarrow C\boldsymbol{\mu} = \begin{pmatrix} 1 & -1 & 3 & 0 & \dots & 0 \end{pmatrix} \begin{pmatrix} \mu_1 \\ \vdots \\ \mu_p \end{pmatrix} = \mu_1 - \mu_2 + 3\mu_3 \\ \Rightarrow W(\theta) &= \frac{n-1}{1} [(\bar{X}_1 - \bar{X}_2 + 3\bar{X}_3) - (\mu_1 - \mu_2 + 3\mu_3)]^2 \frac{1}{C S^2_n C^T} \end{aligned} \]

Note que \[ \begin{aligned} &\begin{pmatrix} 1 & -1 & 3 & 0 & \dots & 0 \end{pmatrix} \begin{bmatrix} S^2_{n,(1,1)} & \dots & S^2_{n,(1,p)} \\ \vdots & \ddots & \vdots \\ S^2_{n,(p,1)} & \dots & S^2_{n,(p,p)} \end{bmatrix} \begin{pmatrix} 1 \\ -1 \\ 3 \\ 0 \\ \dots \\ 0 \end{pmatrix}\\ \Rightarrow& (S^2_{n,(1,1)} - S^2_{n,(2,1)} + 3 S^2_{n,(3,1)}) - (S^2_{n,(1,1)} - S^2_{n,(2,1)} + 3 S^2_{n,(3,1)})\\ &+ 3(S^2_{n,(1,1)} - S^2_{n,(2,1)} + 3 S^2_{n,(3,1)}) \\ & = S^2_{n,(1,1)} + S^2_{n,(2,2)} + 9 S^2_{n,(3,3)} - 2 S^2_{n,(2,1)} + 6 S^2_{n,(1,3)} - 6 S^2_{n,(2,3)} \\ \Rightarrow W(\theta) &= \frac{n-1 [(\bar{X}_1 - \bar{X}_2 + 3\bar{X}_3) - (\mu_1 - \mu_2 + 3\mu_3)]^2}{ S^2_{n,(1,1)} + S^2_{n,(2,2)} + 9 S^2_{n,(3,3)} - 2 S^2_{n,(2,1)} + 6 S^2_{n,(1,3)} - 6 S^2_{n,(2,3)}} \end{aligned} \]

Logo, \[ \begin{aligned} \mathrm{RC}(\mu_1 - \mu_2 + 3\mu_3, \gamma) &= \left\{ \mu_1 - \mu_2 + 3\mu_3 \in \mathbb{R} : W(\theta) \leq q_\gamma \right\} \\ \Rightarrow \mathrm{RC}(\mu_1 - \mu_2 + 3\mu_3, \gamma) &= (\bar{X}_1 - \bar{X}_2 + 3\bar{X}_3) \\ &\pm \sqrt{q_{\gamma} \frac{S^2_{n,(1,1)} + S^2_{n,(2,2)} + 9 S^2_{n,(3,3)} - 2 S^2_{n,(2,1)} + 6 S^2_{n,(1,3)} - 6 S^2_{n,(2,3)}}{n-1}} \end{aligned} \]

Intervalos de confiança são um caso particular

Se \(s=1\), então a região de confiança é simplesmente um IC.

\[ \mathrm{RC}(g(\theta), \gamma) = C \bar{X} \pm \sqrt{q_\gamma \frac{C S^2_n C^T}{n-1}} \]

Forma geral

Para \(s \geq 2\), representamos a região de confiança pela sua definição

\[ \mathrm{RC}(C \boldsymbol{\mu}, \gamma) = \left\{ C \boldsymbol{\mu} : \frac{n-s}{s}(C\bar{X} - C\boldsymbol{\mu})^T [C S^2_n C^T]^{-1} (C\bar{X} - C\boldsymbol{\mu})^T \leq q_{\gamma}\right\} \]

em que \(P(F_{(s, n-s)} \leq q_\gamma) = \gamma\)

43.1.1 Exemplo

Seja \[ g(\theta) = \begin{pmatrix} \mu_1 - \mu_2 \\ \mu_2 - \mu_3 \end{pmatrix} \Rightarrow C = \begin{pmatrix} 1 & -1 & 0 & 0 & \dots & 0 \\ 0 & 1 & -1 & 0 & \dots & 0 \end{pmatrix} \Rightarrow s = 2 \]

Portanto, \[ \mathrm{RC}(g(\theta), \gamma) = \left\{ \begin{pmatrix} \mu_1 - \mu_2 \\ \mu_2 - \mu_3 \end{pmatrix} \in \mathbb{R}^2 : \frac{n-2}{2} \begin{pmatrix} \bar{X}_1 - \bar{X}_2 \\ \bar{X}_2 - \bar{X}_3 \end{pmatrix}^T [C S^2_n C^T]^2 \begin{pmatrix} \bar{X}_1 - \bar{X}_2 \\ \bar{X}_2 - \bar{X}_3 \end{pmatrix} \right\} \]