An Efficient Adaptive Sequential Procedure for Simple Hypotheses with Expression for Finite Number of Applications of Less Effective Treatment
Sampurna Kundu, Jayant Jha, Subir Kumar Bhandari
TL;DR
The paper develops an adaptive sequential testing framework for two simple hypotheses that enforces finite exposure to the inferior treatment by employing a likelihood-ratio–driven allocation that concentrates samples on the better-performing population. It derives an explicit large-sample expression for the expected number of inferior allocations, $E(N_{1,n}) \approx \frac{1}{2}\left(\frac{\sigma_x^{2}}{\eta_x^{2}} + \frac{\sigma_y^{2}}{\eta_y^{2}}\right)$, and proves that this count is finite with all moments finite, while preserving asymptotic efficiency comparable to the $SPRT$ in terms of the average sample number (ASN). The method uses adaptive SPRT with thresholds $a=\log\left(\frac{1-\beta}{\alpha}\right)$ and $b=\log\left(\frac{\beta}{1-\alpha}\right)$ and a stopping rule based on $L=\sum_{i=1}^{n_{max}}\log\left(\frac{f_1(U_i)}{f_0(U_i)}\right)$, ensuring controlled probability of incorrect selection (PICS). Simulations across Normal, Poisson, and Asymmetric Laplace distributions show high PCS and substantially reduced inferior allocations relative to classical SPRT, validating both theoretical findings and practical applicability in ethically sensitive sequential testing.
Abstract
We propose an adaptive sequential framework for testing two simple hypotheses that analytically ensures finite exposure to the less effective treatment. Our proposed procedure employs a likelihood ratio-driven adaptive allocation rule, dynamically concentrating sampling effort on the superior population while preserving asymptotic efficiency (in terms of average sample number) comparable to the Sequential Probability Ratio Test (SPRT). The foremost contribution of this work is the derivation of an explicit closed-form expression for the expected number of applications to the inferior treatment. This approach achieves a balanced method between statistical precision and ethical responsibility, aligning inferential reliability with patient safety. Extensive simulation studies substantiate the theoretical results, confirming stability in allocation and consistently high probability of correct selection (PCS) across different settings. In addition, we demonstrate how the adaptive procedure markedly reduces inferior allocations compared with the classical SPRT, highlighting its practical advantage in ethically sensitive sequential testing scenarios. The proposed design thus offers an ethically efficient and computationally tractable framework for adaptive sequential decision-making.