Confidence Intervals

• Definition

  A level $1-\alpha$ confidence interval for parameter $\theta$ is an interval $[\hat\theta_L, \hat\theta_U]$, where $\hat\theta_L$ and $\hat\theta_U$ are found from data s.t.

  $$ P_\theta(\hat\theta_L \leq \theta \leq \hat\theta_U) = 1-\alpha $$

Pivotal Mothod

• A pivotal quantity (pivot) is a function of the sample measurements and unknown parameter $\theta$, ($\theta$ is the only unknown parameter) and its probability distribution does not depend on $\theta$.

• If $g(x_1,...,x_n;\theta)$ is pivot, Find $c_1$ and $c_2$ s.t.

  $$ P_\theta(c_1 \leq g(X_1,...,X_n) \leq c_2) = 1-\alpha $$

  ⇒ Restate in the form of $P_\theta(\hat\theta_L \leq \theta \leq \theta_U) = 1-\alpha$

Fundamental Sampling Theorem

• $X_1,...,X_n \overset{iid}{\sim} N(\mu, \sigma^2)$
  • $\bar X \sim N(\mu, \sigma^2/n)$
  • $S^2 = \frac{1}{n-1}\sum(X_i-\bar X)^2$ ㅛ $\bar X$
  • $(n-1)S^2/\sigma^2 \sim \chi^2(n-1)$

Approximate confidence intervals

• Slutsky’s Theorem

  $U_n \overset{D}{\to} U$ and $W_n \overset{p}{\to} 1 \Rightarrow U_n/W_n \overset{D}{\to} U$

  • $W_n \overset{D}{\to} N(0,1) \Rightarrow W_n^2 \overset{D}{\to} \chi^2(1)$
  • $T_n \sim t(n), \text{ then } T_n \overset{D}{\to} N(0,1)$ ( $\chi^2(1)/n \overset{p}{\to} 1$ by WLLN)