📂Mathematical Statistics

Interval Estimator

Definition ¹

For a parameter $\theta \in \mathbb{R}$, the ordered pair $\left( L \left( x_{1} , \cdots , x_{n} \right), U \left( x_{1} , \cdots , x_{n} \right) \right)$ is called an Interval Estimate if it satisfies $L \left( \mathbf{x} \right) \le U \left( \mathbf{x} \right)$ for all $\mathbf{x} \in \mathcal{X}$. The random interval $\left[ L \left( \mathbf{X} \right), U \left( \mathbf{X} \right) \right]$ is referred to as an Interval Estimator.

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• $\mathcal{X}$ is the support of $L$ and $U$.

Explanation

Once data $\mathbf{X} = \mathbf{x}$ is observed, statistical inference is made based on whether $L \left( \mathbf{x} \right) \le \theta \le U \left( \mathbf{x} \right)$ is satisfied or not.