Finite-Length Bounds on Hypothesis Testing Subject to Vanishing Type I Error Restrictions

• Published in: IEEE signal processing letters Vol. 28; pp. 229 - 233
• Main Authors: Espinosa, Sebastian, Silva, Jorge F., Piantanida, Pablo
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Computer Science; concentration inequalities; Constrictions; Context; Convergence; Error analysis; Error exponent; Error probability; Errors; False alarms; finite-length analysis; Hypotheses; Hypothesis testing; Lower bounds; Numerical analysis; performance bounds; Probability; Restrictions; Upper bound; Hypothesis testing performance bounds finitelength analysis error exponent concentration inequalities; Hypothesis testing; concentration inequalities; performance bounds; error exponent; finitelength analysis
• ISSN: 1070-9908; 1558-2361
• DOI: 10.1109/LSP.2021.3050381

A central problem in Binary Hypothesis Testing (BHT) is to determine the optimal tradeoff between the Type I error (referred to as false alarm) and Type II (referred to as miss) error. In this context, the exponential rate of convergence of the optimal miss error probability - as the sample size tends to infinity - given some (positive) restrictions on the false alarm probabilities is a fundamental question to address in theory. Considering the more realistic context of a BHT with a finite number of observations, this letter presents a new non-asymptotic result for the scenario with monotonic (sub-exponential decreasing) restriction on the Type I error probability, which extends the result presented by Strassen in 2009. Building on the use of concentration inequalities, we offer new upper and lower bounds to the optimal Type II error probability for the case of finite observations. Finally, the derived bounds are evaluated and interpreted numerically (as a function of the number samples) for some vanishing Type I error restrictions.