Confidence Interval

Definition

Let

• $X$ : Random variable
• $X_1,\dots ,X_n$ : Random sample of $X$, with
  • pdf $f(x;\theta ),\theta \in \Omega$
• $0<\alpha <1$
• $L=L(X_1,\dots ,X_n)$ : Statistic
• $U=U(X_1,\dots ,X_n)$ : Statistic

If $1-\alpha =P_{\theta }[\theta \in (L,U)]$

Then

• We say interval $(L,U)$ is a $(1-\alpha )100\%$ confidence interval for $\theta$
• The probability that the interval $(L,U)$ includes $\theta$ is $1-\alpha$, which is called the confidence coefficient/level of the interval.
• If $E_{\theta }(U_1-L_1)\le E_{\theta }(U_2-L_2),\forall \theta \in \Omega$, then we ay $(L_1,U_1)$ is more efficient than $(L_2,U_2)$

Remark

Once the sample is drawn, the realized confidence intervals are $(l,u)$, where $l,u\in \mathbb{R}$.

If one computes, say, $M$ independent $(1-\alpha )100\%$ confidence intervals, the one would expect to have $(1-\alpha )M$ successful confidence interval, i.e., intervals that contains $\theta$. Hence one feeels $(1-\alpha )100\%$ confident that the true value of $\theta$ lies in $(l,u)$

See also

• Common Confidence Intervals