Statistical inference

Definition

Let $\mathcal{P} = \set{P_\theta : \theta \in \Theta}$ be a statistical model on a sample space $S$. Let $x \in S$ be an observed value of a random element $X$ whose law is $P_\theta$ for an unknown $\theta$. Statistical inference is the derivation, from $x$, of statements about $\theta$ or about a function $\tau(\theta)$. A rule of inference is a map defined on $S$ that does not depend on $\theta$; its main kinds are these.

1. A point estimate: a value $\hat{\tau}(x)$ of an estimator of $\tau(\theta)$.
2. A set estimate: a set $C(x)$ of plausible values of $\tau(\theta)$.
3. A test: a decision $\varphi(x) \in \set{0, 1}$ between a hypothesis $\theta \in \Theta_0$ and its alternative $\theta \in \Theta \setminus \Theta_0$.