Locally most powerful sequential tests of a simple hypothesis vs one-sided alternatives

TL;DR

This paper characterizes the structure of locally most powerful sequential tests for a simple hypothesis against one-sided alternatives in a stochastic process, optimizing the derivative of the power function under error and sample size constraints.

Contribution

It provides a detailed characterization of the structure of optimal sequential tests for one-sided hypothesis testing with constraints on error probability and average sample size.

Findings

Derived the structure of optimal sequential tests.

Maximized the derivative of the power function at the null hypothesis.

Established conditions under which tests are optimal.

Abstract

Let $X_1,X_2,...$ be a discrete-time stochastic process with a distribution $P_\theta$, $\theta\in\Theta$, where $\Theta$ is an open subset of the real line. We consider the problem of testing a simple hypothesis $H_0:$ $\theta=\theta_0$ versus a composite alternative $H_1:$ $\theta>\theta_0$, where $\theta_0\in\Theta$ is some fixed point. The main goal of this article is to characterize the structure of locally most powerful sequential tests in this problem. For any sequential test $(\psi,\phi)$ with a (randomized) stopping rule $\psi$ and a (randomized) decision rule $\phi$ let $\alpha(\psi,\phi)$ be the type I error probability, $\dot \beta_0(\psi,\phi)$ the derivative, at $\theta=\theta_0$, of the power function, and $\mathscr N(\psi)$ an average sample number of the test $(\psi,\phi)$. Then we are concerned with the problem of maximizing $\dot \beta_0(\psi,\phi)$ in the class of all sequential tests such that $$ \alpha(\psi,\phi)\leq \alpha\quad{and}\quad \mathscr N(\psi)\leq \mathscr N, $$ where $\alpha\in[0,1]$ and $\mathscr N\geq 1$ are some restrictions. It is supposed that $\mathscr N(\psi)$ is calculated under some fixed (not necessarily coinciding with one of $P_\theta$) distribution of the process $X_1,X_2...$. The structure of optimal sequential tests is characterized.