In the first essay, we propose ranked means identification and estimation problem, which is a generalized version of the combination of best arm identification (or ranking and selection) and maximum mean estimation. This problem is important with applications in the field of both operation research and electrical engineering. We propose a greedy policy to tackle it. We find that these algorithms can be easily implemented and work well with a couple of estimators. To investigate the performance of the proposed method, we show its nice properties including consistency, asymptotically unbiasedness, and provide finite-time analysis such as probability of misidentification and mean square error. Moreover, the numerical study demonstrates that the numerical results are consistent with the theoretical analysis.

In the second essay, we study the estimation of the probability distribution function of a conditional expectation that cannot be simulated directly but requires estimation via nested simulation. We propose an adaptive nested simulation procedure that determines the number of inner-level observations for each outer-level scenario in an adaptive manner, thus resulting in saving of simulation budget. We show that as the expected simulation budget increases, the proposed procedure leads to an efficient estimator with mean squared error decaying at a rate faster than that of the standard nested simulation procedure.

In the third essay, we study expected exceed loss, value-at-risk and conditional value-at-risk in risk measurement. We propose an adaptive nested simulation procedure based on elimination. In the nested simulation setting, we first need to generate some scenarios and then allocate samples to these scenarios. The procedure gradually eliminated scenarios that are unlikely to make contributions to the risk measurement, and finally most parts of simulation budget is allocated to the target scenarios achieving the goal of budget saving.