Sequential analysis

Definition

Let $X_1, X_2, \dots$ be random variables observed one at a time, and let $\mathcal{F}_n = \sigma(X_1, \dots, X_n)$ be the information after $n$ observations, with $\mathcal{F}_0$ trivial. Let $D$ be a set of possible decisions, such as "accept $H_0$" and "reject $H_0$". A sequential procedure is a pair $(N, \delta)$ such that:

1. $N$ is a stopping time with respect to $(\mathcal{F}_n)_{n \ge 0}$, called the stopping rule.
2. $\delta$ is a terminal decision rule: on the event $\set{N = n}$, the decision $\delta$ is a function of $(X_1, \dots, X_n)$ with values in $D$.

Sequential analysis is the study of sequential procedures, in particular of their error probabilities and of the distribution of the sample size $N$.