Motivated by practical applications in clinical trials and online platforms, we study A/B testing with the aim of estimating a confidence interval (CI) for the average treatment effect (ATE) using the minimum expected sample size. This CI should have a width at most $\epsilon$ while ensuring that the probability of the CI not containing the true ATE is at most $\delta$. To answer this, we first establish a lower bound on the expected sample size needed for any adaptive policy which constructs a CI of ATE with desired properties. Specifically, we prove that the lower bound is based on the solution to a max-min non-convex optimization problem for small $\delta$. Tailoring the “plug-in” approach for the ATE problem, we construct an adaptive policy that is asymptotically optimal, i.e., matches the lower bound on the expected sample size for small $\delta$. Interestingly, we find that, for small $\epsilon$ and $\delta$, the asymptotically optimal fraction of treatment assignment for A and B is proportional to the standard deviation of the outcome distributions of treatments A and B, respectively. However, as the proposed approach can be computationally intensive, we propose an alternative adaptive policy. This new policy, informed by insights from our lower bound analysis, is computationally efficient while remaining asymptotically optimal for small values of $\epsilon$ and $\delta$. Numerical comparisons demonstrate that both policies perform similarly across practical values of $\epsilon$ and $\delta$, offering efficient solutions for A/B testing.

Abstract Face Recognition is a challenging task for recognizing and detecting the identity of an individual. Although, plethora of work has already been done in the field of pattern recognition still there has been lot which has not been addressed in any of the literature. In the current research, we have presented a comparative analysis using three popularly known techniques for face recognition namely, Principal Components Analysis (PCA) using Eigen Faces, Hidden Markov Model (HMM) using Singular Value Decomposition, and Artificial Neural Network (ANN) using Gabor filters. These techniques are implemented and evaluated using various measuring metrics such as false acceptance, false recognition rate, and so on. We used ORL and Yale Face dataset to test the robustness of implemented algorithms. Results show that ANN model for face recognition outperforms the other two techniques by achieving more accurate results and shows the highest recognition rate of 97.49% on ORL database. Moreover, it is also observed that ANN model shows the minimum error count of about 2.502% on ORL database while it is 3.5% on Yale Face dataset. To evaluate further, the implemented techniques are compared with best known techniques in class implemented by various researchers.