Nearly Optimum Properties of Certain Multi-Decision Sequential Rules for General Non-i.i.d. Stochastic Models

TL;DR

This paper develops three multi-hypothesis sequential testing procedures that are nearly optimal in both i.i.d. and non-i.i.d. stochastic models, extending Lai's foundational work to more general settings.

Contribution

It introduces new multi-hypothesis sequential tests that achieve near first-order asymptotic optimality in broad non-i.i.d. models, generalizing previous two-hypothesis results.

Findings

Tests are proven to be asymptotically optimal for small error probabilities.

The methods extend Lai's 1981 results to multiple hypotheses and non-i.i.d. models.

Log-likelihood ratios converge r-completely to positive finite numbers.

Abstract

Dedicated to the memory of Professor Tze Leung Lai, this paper introduces three multi-hypothesis sequential tests. These tests are derived from one-sided versions of the sequential probability ratio test and its modifications. They are proven to be first-order asymptotically optimal for testing simple and parametric composite hypotheses when error probabilities are small. These tests exhibit near optimality properties not only in classical i.i.d. observation models but also in general non-i.i.d. models, provided that the log-likelihood ratios between hypotheses converge r-completely to positive and finite numbers. These findings extend the seminal work of Lai (1981) on two hypotheses.