Deriving Confidence Intervals using Pivots

Published

April 10, 2026

Modified

Invalid Date

1 Notations

  • \(\theta\): parameter of random variable \(X\)
  • \(\hat{\theta} = \hat{\theta}\left(X_1, \ldots, X_n\right)\): estimate of paremeter \(\theta\)
  • \(F_{\theta}\left(x\right)=\mathbb{P}\left(X\leq x\right)\): CDF of r.v. \(X\), i.e, \(X \sim F_{\theta}\)
  • \(q_{p} = F_{\theta}^{-1}\left(p\right)\): p-quantile of \(F_{\theta}\)
    • \(\iff p = \mathbb{P}\left(X \leq q_{p}\right)\)
  • \(\mathrm{SE} := \mathrm{SD}\left(\hat{\theta}\right)\): standard error of estimate \(\hat{\theta}\)
Note

If the argument of SD is an estimate, then we call SD as SE.

2 Confidence Interval (CI)

2.1 Pivot (Pivotal quantity)

Pivot \(T\) is a statistic which follows below conditions:

  1. \(T\) includes the unknown parameter \(\theta\) of randomvariable \(X\)
  2. Its distribution \(F_T\) does not depend on \(\theta\)
  3. \(F_T\) is known distribution.

2.2 Derive CI

Set the pivot \(T = \frac{\hat{\theta}-\theta}{\mathrm{SE}\left(\hat{\theta}\right) } \sim F_T\).
It does not need to be unbiased. In general, \(\mathbb{E}\left(\hat{\theta}\right) \neq \theta\). (Thus the pivot does not be standardized)

Then, followings are hold.

\[ \begin{align*} P\left( q_{\alpha/2} \le T \le q_{1-\alpha/2} \right) &= F_T\left(q_{1-\alpha/2}\right) - F_T\left(q_{\alpha/2}\right) \\ &= 1 - \alpha \end{align*} \]

Apply \(T = \frac{\hat{\theta}-\theta}{\mathrm{SE}}\),

\[ \mathbb{P}\left( q_{\alpha/2} \leq \frac{\hat{\theta} - \theta}{\mathrm{SE}\left(\hat{\theta}\right) } \leq q_{1-\alpha/2}\right) = 1 - \alpha \]

\[ \mathbb{P}\left( \hat{\theta} - q_{1-\alpha/2}\cdot \mathrm{SE}\left(\hat{\theta}\right) \leq \theta \leq \hat{\theta} - q_{\alpha/2}\cdot \mathrm{SE}\left(\hat{\theta}\right) \right) = 1 - \alpha \]

2.3 Conclusion (CI)

The confidence interval of parameter \(\theta\) is:

\[ \theta \in \left(\hat{\theta} - q_{1-\alpha/2}\cdot \mathrm{SE}\left(\hat{\theta}\right) , \quad \hat{\theta} - q_{\alpha/2}\cdot \mathrm{SE}\left(\hat{\theta}\right) \right) \]

2.4 Examples

2.4.1 Symmetry pivot:

If the pivot distribution is symetry, i.e, \(q_{\alpha/2} = -q_{1-\alpha/2}\) for any \(\alpha\).
Then the CI can be simplified as below:

\[ \theta \in \left(\hat{\theta} - q_{1-\alpha/2}\cdot \mathrm{SE}\left(\hat{\theta}\right) , \quad \hat{\theta} + q_{1-\alpha/2}\cdot \mathrm{SE}\left(\hat{\theta}\right) \right) \]

Simply, the CI is:

\[ \hat{\theta} \pm q_{1-\alpha/2}\cdot \mathrm{SE}\left(\hat{\theta}\right) \]

2.4.2 Normal distribution:

If the estimate \(\hat{\theta}\) follows normal distribution, the pivot can be calculated as below:

\[ \begin{align*} T &= \frac{\hat{\theta}-\theta}{\mathrm{SE}\left(\hat{\theta}\right)} \\ &= \frac{\hat{\theta}-\mathbb{E}\left(\hat{\theta}\right)}{\mathrm{SD}\left(\hat{\theta}\right)} \sim Z\left(0,1\right) \end{align*} \]

Thus we can define standard normal distribution \(Z\)’s quantile: \[z_{p} = \Phi^{-1}\left(p\right)\] where \(\Phi\): CDF of standard normal distribution

Simply, we take CI as below: \[ \hat{\theta} \pm z_{1-\alpha/2}\cdot \mathrm{SE}\left(\hat{\theta}\right) \]